tom robbins ww prob lib1 webwork, version 1.7 - demo course

Tom Robbins WW Prob Lib1 WeBWorK, Version 1.7 - Demo CourseWeBWorK problems.

1.(1 pt) Find the value of the permutation:P�7 � 4 ��2.(1 pt) Find the value of the combination:

C�8 � 3 ��3.(1 pt) How many 5-digit numbers can be formed using the

digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9? Repeated digits are allowed.

4.(1 pt) How many different 8-letter words (real or imagi-nary) can be formed from the letters in the word CALCULUS?

5.(1 pt) Determine the size of the sample space that corre-sponds to the experiment of tossing a coin the following numberof times:

(a) 2 timesanswer:(b) 7 timesanswer:(c) n timesanswer:6.(1 pt) An experiment consists of choosing objects without

regards to order. Determine the size of the sample space whenyou choose the following:

(a) 5 objects from 19answer :(b) 2 objects from 9answer :(c) 6 objects from 21answer :7.(1 pt) Suppose you are managing 16 employees, and you

need to form three teams to work on different projects. Assumethat all employees will work on a team, and that each employeehas the same qualifications/skills so that everyone has the sameprobability of getting choosen. In how many different ways canthe teams be chosen so taht the number of employees on eachproject are as follows:

6 � 4 � 6answer :

8.(1 pt) A computer retail store has 8 personal computers instock. A buyer wants to purchase 2 of them. Unknown to eitherthe retail store or the buyer, 2 of the computers in stock havedefective hard drives. Assume that the computers are selected atrandom.

(a) In how many different ways can the 2 computers be cho-sen?

answer:(b) What is the probability that exactly one of the computers

will be defective?answer:(c) What is the probability that at least one of the computers

selected is defective?answer:

9.(1 pt) In how many ways can 5 novels, 4 mathematicsbooks, and 1 biology book be arranged on a bookshelf if

(a) the books can be arranged in any order?answer:(b) the mathematics books must be together and the novels

must be together?answer:(c) the mathematics books must be together but the other

books can be arranged in any order?answer:

10.(1 pt) From a group of 7 women and 9 men a committeeconsisting of 3 men and 4 women is to be formed. How manydifferent committees are possible if

(a) 2 of the men refuse to serve together?answer:(a) 2 of the women refuse to serve together?answer:(a) 1 man and 1 woman refuse to serve together?answer:

Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c�

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1.(1 pt) Evaluate the binomial coefficient: � 139 �

2.(1 pt) Expand the expression using the Binomial Theorem:�4x � 3 � 5 � x5 x4 x3 x2 x 3.(1 pt) Find the coefficient of x7 in the expansion of

3x2 � 5x � 5


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1.(1 pt)The sample space for an experiment contains five sample

points. The probabilities of the sample points are:P�1 �� P

�2 �� 0 � 15

P�3 �� P

�4 �� 0 � 05

P�5 �� 0 � 6

Find the probability of each of the following events:A : Either 4 or 5 occurs �B : Either 4, 3, or 2 occurs �C : 1 does not occur �P�A �� P

�B �� P

�C ��

2.(1 pt)Two fair dice are tossed, and the up face on each die is

recorded. Find the probability of observing each of the follow-ing events:

A : A 6 appears on each of the two dice �B : The sum of the numbers is odd �C : The sum of the numbers is 10 or more �P�A �� P

�B �� P

�C ��

3.(1 pt)Consider the experiment composed of one roll of a fair die

followed by one toss of a fair coin. Determine the probability ofeach of the following events.

A : An odd number appears on the die. �B : An odd number appears on the die; an H appears on the

coin. �C : An H appears on the coin. �P�A �� P

�B �� P

�C ��

4.(1 pt) In the game Roulette, a ball spins on a circular wheelthat is divided into 38 arcs of equal lenght, numbered 00, 0, 1,2, ... , 35, 36. The number on the arc on which the ball stopsis the outcome of one play of the game. The numbers are alsocolored as follows:1, 3, 5, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 36 arered,2, 4, 6, 8, 10, 11, 13, 15, 17, 20, 22, 24, 26, 28, 29, 31, 33, 35are black,0, 00 are green

Define the following events:A : Outcome is an even number (0 and 00 are considered nei-ther odd nor even) �B : Outcome is a red number �C : Outcome is a green number �D : Outcome is a low number (1-18) �

Find the following probablilities:(a) P

�A ��

(b) P�A � B ��

(c) P�B � C � D ��

5.(1 pt)

A couple decided to have 5 children.(a) What is the probability that they will have at least two

boys?(b) What is the probability that all the children will be of the

same gender?

6.(1 pt)Consider two people being randomly selected. (For simplic-

ity, ignore leap years.)(a) What is the probability that two people have a birthday

on the same day of the same month? (not including the year)answer:(b) What is the probability that two people born in April have

a birthday in the first half of the month?answer:7.(1 pt)In a study by the Department of Transportation, there were

a total of 88 drivers that were pulled over for speeding. Out ofthose 88 drivers, 39 were men who were ticketed, 12 were menwho were not ticketed, 8 were women who were ticketed, and29 were women who were not ticketed. Suppose one person waschosen at random.

(a) What is the probability that the selected person is a manwho was not ticketed?

answer:(b) What is the probability that the selected person is a

woman who was ticketed?answer:8.(1 pt) An elementary school is offering 3 language classes:

one in Spanish, one in French, and one in German. Theseclasses are open to any of the 98 students in the school. Thereare 31 in the Spanish class, 38 in the French class, and 20 in theGerman class. There are 13 students that in both Spanish andFrench, 5 are in both Spanish and German, and 8 are in bothFrench and German. In addition, there are 2 students taking all3 classes.

If one student is chosen randomly, what is the probability thathe or she is taking at least one language class?

If two students are chosen randomly, what is the probabilitythat at least one of them is taking German?

9.(1 pt) A group of kids containing 19 boys and 12 girls islined up in random order - that is, each of the 31! permutationsis assumed to be equally likely. What is the probability that theperson in the 12-th position is a girl?

10.(1 pt) An instructor gives his class a set of 18 problemswith the information that the next quiz will consist of a randomselection of 5 of them. If a student has figured out how to do 6 ofthe problems, what is the probability the he or she will answercorrectly

(a) all 5 problems?(b) at least 4 problems?

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11.(1 pt) How many people have to be in a room in order thatthe probability that at least two of them celebrate their birthdayon the same day is at least 0.11? (Ignore leap years, and assumethat all outcomes are equally likely.)

12.(1 pt)A fair coin is tossed three times and the events A, B, and C

are defined as follows:A : At least one head is observed �B : At least two heads are observed �C : The number of heads observed is odd �Find the following probabilities by summing the probabili-

ties of the appropriate sample points:(a) P

�Cc ��

(b) P�A � C ��

(c) P�A � B � C ��

13.(1 pt)A fair coin is tossed three times and the events A, B, and C

are defined as follows:A : At least one head is observed �B : At least two heads are observed �C : The number of heads observed is odd �Find the following probabilities by summing the probabili-

ties of the appropriate sample points:(a) P

�B ��

(b) P�A � B ��

(c) P�A � B � C ��

14.(1 pt) A sample space contains 7 sample points and eventsA and B as seen in the Venn diagram.

Let P�1 �� P

�2 �� P

�3 �� P

�7 �� 0 � 05

P�4 �� P

�5 �� 0 � 15

and P�6 �� 0 � 5.

Use the Venn diagram and the probabilities of the samplepoints to find:

(a) P�Ac ��

(b) P�A � B ��

(c) P�Bc ��

(d) P�A � Ac ��

15.(1 pt) A sample space contains 7 sample points and eventsA and B as seen in the Venn diagram.

Let P�1 �� P

�2 �� P

�3 �� P

�7 �� 0 � 05

P�4 �� P

�5 �� 0 � 15

and P�6 �� 0 � 5.


(a) P�B ��

(b) P�B ��

(c) P�A � B ��

(d) P�A ��

16.(1 pt)The number 64 is written as a sum of three natural numbers

64 � a b c

(the triple�a � b � c � is ordered; e.g., the decompositions 64 �

1 1 62 and 64 � 1 62 1 are different.Also, assume that all the decompositions have equal probabil-ity.)

What is the probability that there exists a triangle with sidesa, b, and c?

17.(1 pt) A quick quiz consists of 4 multiple choice prob-lems, each of which has 4 answers, only one of which is correct.If you make random guesses on all 4 problems,

(a) What is the probability that all 4 of your answers areincorrect?

answer:(b) What is the probability that all 4 of your answers are

correct?answer:


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1.(1 pt)If P

�A �� 0 � 3, P

�B �� 0 � 6, and P

�A � B �� 0, then

(a) P�A �B �� and

(b) P�B �A ��

2.(1 pt)A sample space contains six sample points and events A, B, andC as shown in the Venn diagram. The probablities of the samplepoints are P

�1 �� 0 � 2 � P

�2 �� 0 � 55 � P

�3 �� 0 � 05 � P

�4 �� 0 � 05 �

P�5 �� 0 � 05 � P � 6 �� 0 � 1 �

Use the Venn diagram and the probabilities of the sample pointsto find:

(a) P�C ��

(b) P�B �C ��

(c) P�Bc �Cc ��

3.(1 pt)A sample space contains six sample points and events A, B, andC as shown in the Venn diagram. The probablities of the sample

points are P�1 �� 0 � 15 � P

�2 �� 0 � 55 � P � 3 �� 0 � 1 � P

�4 �� 0 � 05 �

P�5 �� 0 � 1 � P � 6 �� 0 � 0499999999999999 �

Use the Venn diagram and the probabilities of the sample pointsto find:

(a) P�B ��

(b) P�A �B ��

(c) P�C �A ��

4.(1 pt)A box contains one yellow, two red, and three green balls. Twoballs are randomly chosen without replacement. Define the fol-lowing events:

A : One of the balls is yellow �B : At least one ball is red �C : Both balls are green �D : Both balls are of the same color �Find the following conditional probabilities:(a) P

�Bc �A ��

(b) P�D �B ��

(c) P�D �Cc ��

5.(1 pt)A box contains one yellow, two red, and three green balls. Twoballs are randomly chosen without replacement. Define the fol-lowing events:

A : One of the balls is yellow �B : At least one ball is red �C : Both balls are green �D : Both balls are of the same color �Find the following conditional probabilities:(a) P

�B �A ��

(b) P�B �D ��

(c) P�D �C ��

6.(1 pt)”Channel One” is an educational television network for whichparticipating secondary schools are equipped with TV sets inevery classroom. It has been found that 50% of secondary schools subscribe to Channel One, where ofthese subscribers 15% never use Channel One while 10%claim to use it more than 5 times per week.

Find the probability that a randomly selected secondayschool subscribes to Channel One and uses it more than 5 timesper week.

answer:

7.(1 pt)Two fair dice, one blue and one red, are tossed, and the up faceon each die is recorded. Define the following events:

E : The difference of the numbers is 3 or more �F : A 6 on the blue die �Find the following probabilities:(a) P

�E ��

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(b) P�F ��

(c) P�E � F ��

(d) P�E �F ��

(e) P�F �E ��

Are events E and F independent?� A. no� B. yes

8.(1 pt)Two fair dice, one blue and one red, are tossed, and the up faceon each die is recorded. Define the following events:

E : The numbers are equal �F : The difference of the numbers is 3 or more �Find the following probabilities:(a) P

�E ��

(b) P�F ��

(c) P�E � F ��

Are events E and F independent?� A. yes� B. no

9.(1 pt)A sample space contains six sample points and events A, B, andC as shown in the Venn diagram. The probablities of the sam-ple points are P

�1 �� 3

12 � P�2 �� 2

12 � P�3 �� 2

12 � P�4 �� 2

12 �P�5 �� 2

12 � P � 6 �� 112 �

Are events A and C mutually exclusive?� A. No� B. Yes


P�A ��

P�B ��

P�A � B ��

Are events A and B independent?� A. No� B. Yes

10.(1 pt) Scoring a hole-in-one is the greatest shot a golfercan make. Once 7 professional golfers each made holes-in-oneon the 7th hole at the same golf course at the same tournament. Ithas been found that the estimated probability of making a hole-in-one is 1

2830 for male professionals. Suppose that a sample of7 professional male golfers is randomly selected.

(a) What is the probability that all of these golfers make ahole-in-one on the 16th hole at the same tournament?

answer:(b) What is the probability that none of these golfers make a

hole-in-one on the 16th hole at the same tournament?answer:11.(1 pt)

For two events A and B, P�A �� 0 � 8 and P

�B �� 0 � 2.

(a) If A and B are independent, thenP�A � B ��

P�A �B ��

P�A � B ��

(b) If A and B are dependent and P�A �B �� 0 � 2, then

P�A � B ��

P�B �A ��

12.(1 pt)If P

�A �� 0 � 4, P

�B �� 0 � 3, and P

�A � B �� 0 � 58, then

P�A � B �� .

(a) Are events A and B independent? (enter YES or NO)(b) Are A and B mutually exclusive? (enter YES or NO)13.(1 pt)The number 61 is written as a sum of three natural numbers

61 � a b c

(the triple�a � b � c � is ordered; e.g., the decompositions 61 �

16 17 28 and 61 � 17 28 16 are different.Also, assume that all the decompositions have equal probabil-ity.)

Given that there exists a triangle with sides a, b, and c, whatis the probability that this triangle is isosceles?

14.(1 pt) What is the probability that at least one of a pair offair dice lands of 5, given that the sum of the dice is 7?

15.(1 pt) In a certain community, 28% of the families own adog, and 25% of the families that own a dog also own a cat. Itis also know that 33% of all the families own a cat.

What is the probability that a randomly selected family ownsa dog?

What is the conditional probability that a randomly selectedfamily owns a dog given that it owns a cat?

16.(1 pt) Urn A has 5 white and 14 red balls. Urn B has 8white and 10 red balls. We flip a fair coin. If the oucome isheads, then a ball from urn A is selected, whereas if the oucomeis tails, then a ball from urn B is selected. Suppose that a whiteball is selected. What is the probability that the coin landedheads?

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17.(1 pt) The probability of the closing of the ith relay in thecircuits shown is given by pi. Let p1 � 0 � 8 � p2 � 0 � 5 � p3 � 0 � 4 �p4 � 0 � 1 � p5 � 0 � 6 � If all relays function independently. whatis the probability that a current flows between A and B for therespective circuits?

(a) P �(b) P �


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1.(1 pt)(a) Count the number of ways to arrange a sample of 2 elementsfrom a population of 10 elements.

answer:(b) If random sampling is to be employed, the probability

that any particular sample will be selected is

2.(1 pt) A financial firm is performing an assessment test andrelies on a random sampling of their accounts. Suppose this firmhas 6269 customer accounts numbered from 0001 to 6269.

One account is to be chosen at random. What is the proba-bility that the selected account number is 3668?

answer:


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1.(1 pt) Determine whether the following are valid probabil-ity distributions or not. Type ”VALID” if it is valid, or type”INVALID” if it is not a valid probability distributions.

(a)

x 2 3 6 8P�x � 0 � 6 0 � 2 0 � 1 0 � 1

answer:(b)P�x �� 1

3x , where x � 1 � 2 � 3 ��answer:(c)

x 2 3 6 8P�x � 0 � 1 0 � 1 � 0 � 3 1 � 1

answer:2.(1 pt) The mean and standard deviation of a random vari-

able x are 10 and 4 respectively. Find the mean and standarddeviation of the given random variables:

(1) y � x 6µ �σ �

(2) v � 4xµ �σ �

(3) w � 4x 6µ �σ �

3.(1 pt)

x 1 2 3 4 5P�x � 0 � 2 0 � 2 0 � 1 0 � 1 0 � 4

Given the discrete probability distribution above, determinethe following:

(a) P�x � 5 ��

(b) P�x � 4 ��

(c) P�x � 4 or x � 2 ��

4.(1 pt) Three dice are rolled. Let the random variable x rep-resent the sum of the 3 dice. By assuming that each of the 63

possible outcomes is equally likely, find the probability that xequals 5.

P�x � 5 ��

5.(1 pt) Let x represent the difference between the number ofheads and the number of tails when a coin is tossed 32 times.Then

P�x � 12 ��

6.(1 pt) Four buses carrying 148 high school students arriveto Montreal. The buses carry, respectively, 34, 48, 27, and 39students. One of the studetns is randomly selected. Let X de-note the number of students that were on the bus carrying this

randomly selected student. One of the 4 bus drivers is also ran-domly selected. Let Y denote the number of students on his bus.Compute the expectations of X and Y :

E�X ��

E�Y ��

7.(1 pt) Four buses carrying 153 high school students arriveto Montreal. The buses carry, respectively, 36, 48, 27, and 42students. One of the studetns is randomly selected. Let X de-note the number of students that were on the bus carrying thisrandomly selected student. One of the 4 bus drivers is also ran-domly selected. Let Y denote the number of students on his bus.Compute the expectations and variances of X and Y :

E�X ��

Var�X � �

E�Y ��

Var�Y ��

8.(1 pt) Two fair dice are rolled 5 times. Let the randomvariable x represent the number of times that the sum 7 occurs.The table below describes the probability distribution. Find thevalue of the missing probability.

x P�x �

0 0.4018775720164611 0.4018775720164612 0.1607510288065843 0.032150205761316945 0.000128600823045267

Would it be unusual to roll a pair of dice 5 times and get no7s?(enter YES or NO)

9.(1 pt) A rock concert producer has scheduled an outdoorconcert. The producer estimates the attendance will depend onthe weather according to the following table.

Weather Attendance Probabilitywet, cold 4000 0 � 1

wet, warm 15000 0 � 3dry, cold 20000 0 � 2

dry, warm 45000 0 � 4(a) What is the expected attendace?answer:(b) If tickets cost $ 15 each, the band will cost $ 200000,

plus $ 40000 for administration. What is the expected profit?answer:

10.(1 pt) Prizes and the chances of winning in a sweepstakesare given in the table below.

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Prize Chances$15,000,000 1 chance in 200,000,000

$550,000 1 chance in 100,000,000$25,000 1 chance in 10,000,000$15,000 1 chance in 5,000,000

$800,000 1 chance in 100,000A watch valued at $45 1 chance in 5,000

(a) Find the expected value (in dollars) of the amount wonby one entry.

(b) Find the expected value (in dollars) if the cost of enteringthis sweepstakes is the cost of a postage stamp (34 cents)


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1.(1 pt) If x is a binomial random variable, compute p�x � for

each of the following cases:(a) n � 5 � x � 3 � p � 0 � 1

p�x ��(b) n � 3 � x � 3 � p � 0 � 1

p�x ��(c) n � 5 � x � 2 � p � 0 � 8

p�x ��(d) n � 3 � x � 0 � p � 0 � 5

p�x ��2.(1 pt) The rates of on-time flights for commercial jets are

continuously tracked by the U.S. Department of Transportation.Recently, Southwest Air had the best reate with 80 % of itsflights arriving on time. A test is conducted by randomly se-lecting 16 Southwest flights and observing whether they arriveon time.

(a) Find the probability that at least 5 flights arrive late.

(b) Would it be unusual for Southwest to have 14 flights arriveon time?(Enter YES or NO)

3.(1 pt) If x is a binomial random variable, compute the mean,the standard deviation, and the variance for each of the follow-ing cases:

(a) n � 6 � p � 0 � 1µ �σ2 �σ �

(b) n � 5 � p � 0 � 5µ �σ2 �σ �

(c) n � 4 � p � 0 � 3µ �σ2 �σ �

(d) n � 4 � p � 0 � 5µ �σ2 �σ �

4.(1 pt) A quiz cosists of 10 multiple-choice questions, eachwith 5 possible answers. For someone who makes randomguesses for all of the answers, find the probability of passingif the minimum passing grade is 40 %.

5.(1 pt) If x is a binomial random variable, compute P�x � for

each of the following cases:(a) P

�x � 1 �!� n � 5 � p � 0 � 7

P�x ��(b) P

�x " 6 �!� n � 8 � p � 0 � 3

P�x ��(c) P

�x � 1 �!� n � 4 � p � 0 � 6

P�x ��(d) P

�x � 1 �!� n � 5 � p � 0 � 5

P�x ��6.(1 pt)The Census Bureau reports that 82% of Americans over the

age of 25 are high school graduates. A survey of randomly se-lected residents of certain county included 1350 who were overthe age of 25, and 1156 of them were high school graduates.

(a) Find the mean and standard deviation for the number ofhigh school graduates in groups of 1350 Americans over the ageof 25.

Mean =Standard deviation =(b) Is that county result of 1156 unusually high, or low, or

neither?(Enter HIGH or LOW or NEITHER)

7.(1 pt) To determine whether or not they have a certain de-sease, 120 people are to have their blood tested. However, ratherthan testing each individual separately, it has been decided firstto group the people in groups of 10. The blood samples of the10 people in each group will be pooled and analized together. Ifthe test is negative. one test will suffice for the 10 people (weare assuming that the pooled test will be positive if and only if atleast one person in the pool has the desease); whereas, if the testis positive each of the 10 people will also be individually testedand, in all, 11 tests will be made on this group. Assume theprobability that a person has the desease is 0.09 for all people,independently of each other, and compute the expected numberof tests necessary for each group.

answer:8.(1 pt) A man claims to have extrasensory perception. As a

test, a fair coin is flipped 30 times, and the man is asked to pre-dict the outcome in advance. He gets 26 out of 30 correct. Whatis the probability that he would have done at least this well if hehad no ESP?


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1.(1 pt) Given that x is a random variable having a Poissondistribution, compute the following:

(a) P�x � 8 � when µ � 1 � 5

P�x ��(b) P

�x � 8 � when µ � 2 � 5

P�x ��(c) P

�x " 3 � when µ � 2

P�x ��(d) P

�x � 1 � when µ � 4 � 5

P�x ��2.(1 pt) A statistics professor finds that when she schedules

an office hour for student help, an average of 3 � 5 students arrive.Find the probability that in a randomly selected office hour, thenumber of student arrivals is 2.

3.(1 pt) The mean number of patients admitted per day tothe emergency room of a small hospital is 0 � 5. If, on any givenday, there are only 3 beds available for new patients, what isthe probability that the hospital will not have enough beds toaccommodate its newly admitted patients?

answer:

4.(1 pt) A certain typing agency employs two typists. Theaverage number of errors per article is 3 � 5 when typed by thefirst typist and 2 when typed by the second. If your article isequally likely to be typed by either typist, find the probabilitythat it will have no errors.


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1.(1 pt) Find the following probabilities for the standard nor-mal random variable z:

(a) P� � 0 � 7 � z � 0 � 53 ��

(b) P� � 1 � 59 � z � 0 � 11 ��

(c) P�z � 1 � 54 ��

(d) P�z "#� 1 � 94 � �

2.(1 pt) Assume that the readings on the thermometers arenormally distributed with a mean of 0 $ and a standard deviationof 1 � 00 $ C. A thermometer is randomly selected and tested. Findthe probability of each reading in degrees.

(a) Between 0 and 0 � 55:(b) Between � 1 � 45 and 0:(c) Between � 2 � 45 and 0 � 75:(b) Less than 1:(c) Greater than � 2 � 17:

3.(1 pt) Find the value of the standard normal random vari-able z, called z0 such that:

(a) P�z � z0 �� 0 � 9741

z0 �(b) P

� � z0 � z � z0 �� 0 � 4616z0 �

(c) P� � z0 � z � z0 �� 0 � 1932

z0 �(d) P

�z � z0 �� 0 � 1129

z0 �(e) P

� � z0 � z � 0 �� 0 � 3264z0 �

(f) P� � 1 � 26 � z � z0 �� 0 � 7100

z0 �4.(1 pt) Assume that the readings on the thermometers are

normally idstributed with a mean of 0 $ and a standard deviationof 1 � 00 $ C.

Find P80, the 80th percentile.This is the temperature reading separating the bottom 80 %

from the top 20 %.

5.(1 pt) Suppose that the readings on the thermometers arenormally distributed with a mean of 0 $ and a standard deviationof 1 � 00 $ C.If 5% of the thermometers are rejected because they have read-ings that are too high, but all other thermometers are acceptable,find the reading that separates the rejected thermometers fromthe others.

6.(1 pt)For a normal distribution, find the percentage of data that are(a) Between µ � 3σ and µ 3σ(b) Greater than 1.5 standard deviations below the mean(c) Less than µ � 3σ

7.(1 pt) Suppose the random variable x is best described by anormal distribution with µ � 21 and σ � 6 � 1. Find the z-scorethat corresponds to each of the following x values.

(a) x � 20z �

(b) x � 11z �

(c) x � 22z �

(d) x � 33z �

(e) x � 18z �

(f) x � 17z �

8.(1 pt) Suppose x is a normally distributed random variablewith µ � 10 � 8 and σ � 3 � 2. Find each of the following probabil-ities:

(a) P�9 � 7 � x � 15 � 6 ��

(b) P�7 � 9 � x � 16 � 4 ��

(c) P�5 � x � 16 � 4 ��

(d) P�x � 7 � 9 ��

(e) P�x � 13 � 4 ��

9.(1 pt) The physical fitness of an athlete is often measuredby how much oxygen the athlete takes in (which is recorded inmilliliters per kilogram, ml/kg). The mean maximum oxygenuptake for elite athletes has been found to be 65 with a standarddeviation of 7 � 9. Assume that the distribution is approximatelynormal.

(a) What is the probability that an elite athlete has a maxi-mum oxygen uptake of at least 50 ml/kg?

answer:(b) What is the probability that an elite athlete has a maxi-

mum oxygen uptake of 55 ml/kg or lower?answer:(c) Consider someone with a maximum oxygen uptake of

39 ml/kg. Is it likely that this person is an elite athlete? Write”YES” or ”NO.”

answer:

10.(1 pt) The combined math and verbal scores for femalestaking the SAT-I test are normally distributed with a mean of 998and a standard deviation of 202 (based on date from the CollegeBoard). If a college includes a minimum score of 875 amongits requirements, what percentage of females do not satisfy thatrequirement?

11.(1 pt) The extract of a plant native to Taiwan has beentested as a possible treatment for Leukemia. One of the chemi-cal compounds produced from the plant was analyzed for a par-ticular collagen. The collagen amount was found to be normally

1

distributed with a mean of 73 and standard deviation of 5 gramsper mililiter.

(a) What is the probability that the amount of collagen isgreater than 63 grams per mililiter?

answer:(b) What is the probability that the amount of collagen is less

than 88 grams per mililiter?answer:(c) What percentage of compounds formed from the extract

of this plant fall within 3 standard deviations of the mean?answer: %12.(1 pt) IQ scores are normally distributed with a mean of

100 and a standard deviation of 15. Mensa is an internationalsociety that has one - and only one - qualification for member-ship: a score in the top 2on an IQ test.

(a) What IQ score should one have in order to be eligible forMensa?

(b) In a typical region of 110,000 people, how many are eli-gible for Mensa?

13.(1 pt) Using diaries for many weeks, a study on thelifestyles of visually impaired students was conducted. The stu-dents kept track of many lifestyle variables including how manyhours of sleep obtained on a typical day. Researchers found thatvisually impaired students averaged 8 � 58 hours of sleep, witha standard deviation of 2 � 6 hours. Assume that the number of

hours of sleep for these visually impaired students is normallydistributed.

(a) What is the probability that a visually impaired studentgets less than 6 hours of sleep?

answer:(b) What is the probability that a visually impaired student

gets between 6 � 7 and 8 � 51 hours of sleep?answer:(c) Fourty percent of students get less than how many hours

of sleep on a typical day?answer: hours14.(1 pt) Women’s weights are normally distributed with a

mean given by µ � 143 lb and a standard deviation given byσ � 29 lb. Find the second decile, D2, which separates the bot-tom 20% from the top 80%.

15.(1 pt) Healty people have body temperatures that are nor-mally distributed with a mean of 98 � 20 $ F and a standard devia-tion of 0 � 62 $ F .

(a) If a healthy person is randomly selected, what is the prob-ability that he or she has a temperature above 98 � 9 $ F?

answer:(b) A hospital wants to select a minimum temperature for

requiring further medical tests. What should that temperaturebe, if we want only 2 � 5% of healty people to exceed it?

answer:


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1.(1 pt) Assume that women’s weights are normally dis-tributed with a mean given by µ � 143 lb and a standard de-viation given by σ � 29 lb.

(a) If 1 woman is randomly selected, find the probabity thather weight is above 174

(b) If 3 women are randomly selected, find the probabilitythat they have a mean weight above 174

(c) If 59 women are randomly selected, find the probabilitythat they have a mean weight above 174

2.(1 pt) Cans of regular Coke are labeled as containing 12 oz.Statistics students weighted the content of 9 randomly chosencans, and found the mean weight to be 12 � 1.

Assume that cans of Coke are filled so that the actual amountsare normally distributed with a mean of 12 � 00 oz and a standarddeviation of 0 � 09 oz. Find the probability that a sample of 9 canswill have a mean amount of at least 12 � 1 oz.

3.(1 pt) Scores for men on the verbal portion of the SAT-Itest are normally distributed with a mean of 509 and a standarddeviation of 112.

(a) If 1 man is randomly selected, find the probability thathis score is at least 585 � 5.

(b) If 12 men are randomly selected, find the probability thattheir mean score is at least 585 � 5.

12 randomly selected men were given a review course be-fore taking the SAT test. If their mean score is 585 � 5, is there astrong evidence to support the claim that the course is actuallyeffective?

(Enter YES or NO)4.(1 pt)The Central Limit Theorem says� A. When n � 30, the sampling distribution of x will be

approximately a normal distribution.� B. When n � 30, the original population will be approx-imately a normal distribution.� C. When n " 30, the sampling distribution of x will beapproximately a normal distribution.

� D. When n " 30, the original population will be approx-imately a normal distribution.� E. None of the above

5.(1 pt) A soft drink bottler purchases glass bottles from avendor. The bottles are required to have an internal pressure ofat least 150 pounds per square inch (psi). A prospective bottlevendor claims that its production process yields bottles with amean internal pressure of 157 psi and a standard deviation of 3psi. The bottler strikes an agreement with the vendor that per-mits the bottler to sample from the production process to verifythe claim. the bottler randomly selects 70 bottles from the last10000 produced, measures the internal pressure of each, andfinds the mean pressure for the sample to be 1 � 1 psi below theprocess mean cited by the vendor.

(a) Assuming that the vendor is correct in his claim, what isthe probability of obtaining a sample mean this far or fartherbelow the process mean?

(b) If the standard deviation were 3 psi as claimed, but themean was 159 psi, what is the probability of obtaining a samplemean of 155 � 9 psi or below?

(c) If the process mean were 157psi as claimed, but the standard deviation was 1 psi, what is theprobability of obtaining a sample mean of 155 � 9 psi or below?

6.(1 pt) Suppose that from the past experience a professorknows that the test score of a student taking his final exami-nation is a random variable with mean 64 and standard devia-tion 11 � How many students would have to take the examinationto ensure, with probability at least 0 � 94, that the class averagewould be within 3 of 64?

7.(1 pt) 90 numbers are rounded off to the nearest integer andthen summed. If the individual round-off error are uniformlydistributed over

� �%� 5 �� 5 � what is the probability that the resul-tant sum differs from the exact sum by more than 5?

8.(1 pt) A die is continuously rolled until the total sum of allrolls exceeds 275 � What is the probability that at least 85 rollsare necessary?


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1.(1 pt) Use normal approximation to estimate the probabil-ity of getting at least 51 girls in 100 births. Assume that boysand girls are equally likely.

2.(1 pt) Use normal approximation to estimate the probabil-ity of passing a true/false test of 50 questions if the minimumpassing grade is 70% and all responses are random guesses.

3.(1 pt) An airline company is considering a new policy ofbooking as many as 298 persons on an airplane that can seat

only 270.5. (Past studies have revealed that only 83% of thebooked passengers actually arrive for the flight.) Estimate theprobability that if the company books 298 persons. not enoughseats will be available.

4.(1 pt) A multiple-choice test consists of 26 questions withpossible answers of a, b, c, d, e, f. Estimate the probability thatwith random quessing, the number of correct answers is at least10.


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1.(1 pt) A manager of an apartment store reports that the timeof a customer on the second floor must wait for the elevator has auniform distribution ranging from 2 to 4 minutes. If it takes theelevator 30 seconds to go from floor to floor, find the probabilitythat a hurried customer can reach the first floor in less than 3 � 25minutes after pushing the elevator button on the second floor.

answer :2.(1 pt) Suppose the time to process a loan application fol-

lows a uniform distribution over the range 8 to 14 days. What isthe probability that a randomly selected loan application takeslonger than 13 days to process?

answer:3.(1 pt) Suppose x is a random variable best described by a

uniform probability that ranges from 2 to 4. Compute the fol-lowing:

(a) the probability density function f�x ��

(b) the mean µ �(c) the standard deviation σ �(d) P

�µ � σ � x � µ σ �&�

(e) P�x � 3 � 43 ��

4.(1 pt) Suppose a random variable x is best described bya uniform probability distribution with range 0 to 4. Find thevalue of a that makes the following probability statements true.

(a) P�x � a �� 0 � 64

a �

(b) P�x � a �� 0 � 46

a �(c) P

�x � a �� 0 � 98

a �(d) P

�x " a �� 0 � 73

a �(e) P

�1 � 56 � x � a �� 0 � 08

a �5.(1 pt) The weather in Rochester in December is fairly con-

stant. Records indicate that the low temperature for each day ofthe month tend to have a uniform distribution over the interval15 $ to 35 $ F. A business man arrives on a randomly selected dayin December.

(a) What is the probability that the temperature will be above19 $ ?answer:

(b) What is the probability that the temperature will be be-tween 18 $ and 33 $ ?answer:

(c) What is the expected temperature?answer:

6.(1 pt) If a is uniformly distributed over '(� 29 � 31 ) , what isthe probability that the roors of the equation

x2 ax a 63 � 0

are both real?


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1.(1 pt) Suppose that the time (in hours) required to repaira machine is an exponentially distributed random variable withparameter λ � 0 � 8. What is

(a) the probability that a repair time exceeds 8 hours?(b) the conditional probability that a repair takes at least 8

hours, given that it takes more than 5 hours?2.(1 pt) Suppose that the life distribution of an item has haz-

ard rate function λ�t �� 3t2 � t " 0. What is the probability that

(a) the item survives to age 3?(b) the item’s lifetime is between 1 and 4?(c) a 0 � 5-year-old item will survive to age 2?

3.(1 pt) Let X be an exponential random variable with param-eter λ � 8, and let Y be the random variable defined by Y � 9eX .Compute the probability density function of Y :

fY�t ��


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1.(1 pt) You’ll need to use the formatted text mode in order todo this problem: click the ”formatted text” button on the bottomof the page and then click ”submit answers”.

Let X be a random variable with probability density function

f�x ��+* c

�4x � x2 � if 0 � x � 4

0 otherwiseFind the value of c:c �Find the cumulative distribution function of X :

F�x ��-,. / if x � 0

if 0 � x � 4if x � 4

2.(1 pt) You’ll need to use the formatted text mode in order todo this problem: click the ”formatted text” button on the bottomof the page and then click ”submit answers”.

The probability density function of X , the lifetime of a cer-tain type of device (measured in months), is given by

f�x ��0* 0 if x � 10

10x2 if x " 10

Find the following:P�X " 44 ��

The cumulative distribution function of X :

F�x �� * if x � 10

if x " 10

The probability that at least one out of 4 devices of this typewill function for at least 19 months:

3.(1 pt) The density function of X is given by

f�x �� * a bx2 if 0 � x � 1

0 otherwise

If the expectation of X is E�X ��1� 3 � 25, find a and b.

a �b �


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1.(1 pt) The joint probability density function of X and Y isgiven by

f�x � y �� c

�y2 � 144x2 � e 2 y �3� y

12� x � y

12� 0 � y � ∞

Find c and the expected value of X :c �E�X ��

2.(1 pt) x and y are uniformly distributed over the interval' 0 � 1 )4� Find the probability that � x � y � , the distance between xand y, is less than 0 � 5 �

3.(1 pt) A man and a woman agree to meet at a cafe aboutnoon. If the man arrives at a time uniformly distributed between11 : 40 and 12 : 10 and if the woman independently arrives ata time uniformly distributed between 11 : 55 and 12 : 40, whatis the probability that the first to arrive waits no longer than 15minutes?

4.(1 pt) Two points are selected randomly on a line of length28 so as to be on opposite sides of the midpoint of the line. Inother words, the two points X and Y are independent randomvariables such that X is uniformly distriuted over ' 0 � 14 � and Yis uniformly distributed over

�14 � 28 ) . Find the probability that

the distance between the two points is greater than 4.answer:5.(1 pt) Let

f�x �� * cx10y5 if 0 � x � 1 � 0 � y � 1

0 otherwiseFind the following:(a) c such that f

�x � y � is a probability density function:

c �(b) Expected values of X and Y :E�X ��

E�Y ��

(c) Are X and Y independent? (enter YES or NO)6.(1 pt) Let A, B, and C be independent random variables,

uniformly distributed over ' 0 � 9 )4�5' 0 � 10 )6� and ' 0 � 1 ) respectively.What is the probability that both roots of the equation Ax2 Bx C � 0 are real?

7.(1 pt) Assume that the monthly worldwide average numberof airplaine crashes of commercial airlines is 2 � 2. What is theprobability that there will be

(a) less than 2 such accidents in the next month?(b) at least 4 such accidents in the next 2 months?(c) exactly 7 such accidents in the next 4 months?8.(1 pt) Andrew’s bowling scores are approximately nor-

mally distributed with mean 110 and standard deviation 21 �

while Pam’s scores are normally distributed with mean 165 andstandard deviation 22 � If Andrew and Pam each bowl one game,then assuming that their scores are independent random vari-ables, approximate the probability that the total of their scoresis above 265 �

9.(1 pt) The joint probability mass function of X and Y isgiven by

p�1 � 1 �� 0 � 15 p

�1 � 2 �� 0 � 15 p

�1 � 3 �� 0 � 1

p�2 � 1 �� 0 � 05 p

�2 � 2 �� 0 p

�2 � 3 �� 0 � 1

p�3 � 1 �� 0 � 1 p

�3 � 2 �� 0 � 1 p

�3 � 3 �� 0 � 25

(a) Compute the conditional mass function of Y given X � 1:P�Y � 1 �X � 1 ��

P�Y � 2 �X � 1 ��

P�Y � 3 �X � 1 ��(b) Are X and Y independent? (enter YES or NO)(c) Compute the following probabilities:P�X Y " 2 ��

P�XY � 4 ��

P� X

Y " 2 ��10.(1 pt) The joint probability mass function of X and Y is

given by

p�1 � 1 �� 0 p

�1 � 2 �� 0 � 15 p

�1 � 3 �� 0 � 1

p�2 � 1 �� 0 � 15 p

�2 � 2 �� 0 � 25 p

�2 � 3 �� 0 � 05

p�3 � 1 �� 0 � 1 p

�3 � 2 �� 0 � 15 p

�3 � 3 �� 0 � 05

Compute the following probabilities:P�X Y " 2 ��

P�XY � 3 ��

P� X

Y " 2 ��11.(1 pt) The joint probability mass function of X and Y is

given by

p�1 � 1 �� 0 � 25 p

�1 � 2 �� 0 � 1 p

�1 � 3 �� 0 � 15

p�2 � 1 �� 0 � 05 p

�2 � 2 �� 0 p

�2 � 3 �� 0 � 15

p�3 � 1 �� 0 � 1 p

�3 � 2 �� 0 � 1 p

�3 � 3 �� 0 � 1

(a) Compute the conditional mass function of Y given X � 1:P�Y � 1 �X � 1 ��

P�Y � 2 �X � 1 ��

P�Y � 3 �X � 1 ��(b) Are X and Y independent? (enter YES or NO)

12.(1 pt) Two points along a straight stick of length 44cm arerandomly selected. The stick is then broken at those two points.Find the probability that all of the resulting pieces have lenghtat least 4cm.

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1.(1 pt) A fair die is rolled 15 times. What is the expectedsum of the 15 rolls?

2.(1 pt) 26 people arrive separately to a professional din-ner. Upon arrival, each person looks to see if he or she hasany friends among those present. That person then either sits atthe table of a friend or at an unoccupied table is none of thosepresent is a friend. Assuming that each of the � 26

2 � pairs of peo-ple are, independently, friends with probability 0 � 6 � find the ex-pected number of occupied tables.

3.(1 pt) Consider n � 10 independent flips of a fair coin. Saythat a changeover occurs whenever an outcome differs from theone preceding it. For example, if n � 6 and the outcome isT H T T H T � then there is a total of 4 changeorvers. Findthe expected number of changeovers for n � 10 �

4.(1 pt) If E 'X )7�1� 1 and Var�X �� 5, then

E ' � 1 2X � 2 )8�andVar

�4 5X � 2 � .


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1.(1 pt) Determine whether the following examples of dataare quantitative or qualitative. Write ”QUANTITATIVE” forquantitative and ”QUALITATIVE” for qualitative. (withoutquotations)

(a) The amount of bacteria on a piece of moldy bread.answer:

(b) The occupation of your neighbors.answer:

(c) The marital status of your coworkers.answer:

(d) Your college GPA.answer:

2.(1 pt) Determine whether the following examples are dis-crete or continuous data sets. Write ”DISCRETE” for discreteand ”CONTINUOUS” for continuous. (without quotations)

(a) The number of errors found on a student’s research paper.answer:

(b) The number of students applying to graduate schools.answer:

(c) The number of voters who vote Democratic.answer:

(d) The length of time it takes to fill up your gas tank.answer:

3.(1 pt) Determine whether the follow descriptions corre-spond to an observational study or an experiment. Write ”EX-PERIMENT” for experiment and ”OBSERVATION” for obser-vational study. (without quotations)

(a) Studying how patients respond when given a placebo.answer:

(b) Classifying different stages of a child’s language devel-opment.answer:

(c) A new antibiotic is tested in effectiveness by recordinghow the drug works on patients that already take the drug.answer:

4.(1 pt)

Grade on Statistics Exam FrequencyBelow 50 250 � 59 260 � 69 770 � 79 1680 � 89 17

90 � 100 15

Given the frequency table above, construct the following:(a) The relative frequency table that corresponds with the

above table.

Grade on Statistics Exam Relative FrequencyBelow 5050 � 5960 � 6970 � 7980 � 89

90 � 100

(b) The cumulative frequency table that corresponds with theabove table.

Grade on Statistics Exam Cumulative FrequencyBelow 5050 � 5960 � 6970 � 7980 � 89

90 � 100

5.(1 pt) Complete the table below.

Books read within the past year Frequency Relative Frequencynone 30 � 4 85 � 9 6

10 � 14 0 � 28846153846153815 � 19 1020 � 25 10

total 52 1


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1.(1 pt) The length (pgs) of math research projects is givenbelow. Using this information, calculate the range, variance, andstandard deviation.

24 � 35 � 28 � 24 � 57 � 50 � 29 � 39 � 42 � 25 � 14

range �variance �standard deviation �

2.(1 pt)Calculate the mode, mean, and median of the following data:

15 � 15 � 22 � 20 � 18 � 14 � 9 � 15

Mode =Mean =Median =3.(1 pt) Given the data set below, calculate the range, vari-

ance, and standard deviation.

49 � 40 � 24 � 11 � 32 � 17 � 21 � 11 � 9

range � variance � standard deviation �4.(1 pt) For each of the given the data sets below, calculate

the mean, variance, and standard deviation.(a) 89 � 85 � 44 � 15 � 37 � 75 � 52 � 49 � 23

mean � variance � standard deviation �(b) 58 � 44 � 48 � 60 � 41 � 48

mean � variance � standard deviation �(c) 2 � 1 � 2 � 3 � 7 � 2 � 3 � 2 � 2

mean � variance � standard deviation �5.(1 pt) Calculate the mean and median of the following

grades on a math test:

95 � 94 � 88 � 85 � 85 � 84 � 84 � 77 � 69 � 64 � 35

Mean =Median =Is this data set skewed to the right, symmetric, or skewed to

the left?(Enter SR, SYM, or SL);

6.(1 pt) If the average low temperature of a winter month inRochester, NY is 13 $ and the standard deviation is 3 � 9, then ac-cording to Chebyshev’s theorem, the percentage of averages lowtemperatures in Rochester, NY between 5 � 2 $ and 20 � 8 $ is%.

7.(1 pt) Calculate the mean and median of the following data:� 15 �� 5 � 5 � 6 � 7

Mean =Median =Is this data set skewed to the right, symmetric, or skewed to

the left?(Enter SR, SYM, or SL.)

8.(1 pt) Find the indicated decile of the following data set

17 � 39 � 20 � 29 � 21 � 50 � 24 � 34 � 45 � 58 � 17 � 10

D2 �9.(1 pt)IQ scores have a mean of 100 and a standard deviation of 15.

Greg has an IQ of 127.What is the difference between Greg’s IQ and the mean?Convert Greg’s IQ score to a z score:

10.(1 pt)Ted took 4 courses last semester: Physics, Spanish, Calculus,

and Biology. The means and standard deviations for the finalexams, and Ted’s scores are given in the table below. ConvertTed’s score into z scores.

Subject Mean Stand. dev. Ted’s score Ted’s z scorePhysics 60 14 60Spanish 44 12 32Calculus 70 12 85Biology 77 10 72

On what exam did Ted have the highest relative score?(Enter the suject.)

11.(1 pt)Here is a list of 25 scores on a Math midterm exam:

38 � 5 � 41 � 5 � 52 � 52 � 5 � 61 � 63 � 63 � 5 � 68 � 69 � 69 �78 � 5 � 79 � 80 � 83 � 87 � 88 � 5 � 88 � 5 � 91 � 91 � 5 � 92 �92 � 5 � 94 � 94 � 97 � 97

Find P78:12.(1 pt)Here is a list of 27 scores on a Statistics midterm exam:

20 � 30 � 31 � 32 � 46 � 48 � 49 � 52 � 54 �59 � 61 � 69 � 71 � 73 � 74 � 79 � 81 � 81 �81 � 85 � 86 � 87 � 88 � 91 � 94 � 96 � 97

Find Q1:


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1.(1 pt) Match the confidence level with the confidence inter-val for µ.

1. x 9 1 � 282 : σ;n <

2. x 9 1 � 645 : σ;n <

3. x 9 1 � 96 : σ;n <

A. 90%B. 95%C. 80%

2.(1 pt) Starting salaries of 70 college graduates who havetaken a statistics course have a mean of $44,842 and a standarddeviation of $8,417.Using a 0.94 degree of confidence, find both of the following:

A. The margin of error E

B. The conficence interval for the mean µ:� µ �3.(1 pt) A random sample of 120 observations produced a

mean of x � 27 � 7 and a standard deviation x � 2 � 6.(a) Find a 95% confidence interval for µ� µ �(b) Find a 99% confidence interval for µ� µ �(c) Find a 90% confidence interval for µ� µ �4.(1 pt) Listed below are the lenths (in minutes) of randomly

selected music CDs. Construct a 96% confidence interval forthe mean length of all such CDs.

55 � 52 64 � 77 63 � 32 38 � 92 66 � 53 33 � 91 57 � 2159 � 8 64 � 76 55 � 41 52 � 64 59 � 52 53 � 4 62 � 4258 � 9 55 � 66 68 � 69 37 � 21 53 � 68 53 � 98 50 � 3851 � 68 55 � 22 82 � 39 57 � 48 58 � 56 59 � 65 56 � 3736 � 98 41 � 82 64 � 28 36 � 85 39 � 58 48 � 62 46 � 3369 � 34 58 � 14 28 � 08 53 � 22 55 � 36� µ �

5.(1 pt) A random sample of n measurements was selectedfrom a population with unknown mean µ and standard deviationσ. Calculate a 90 % confidence interval for µ for each of thefollowing situations:

(a) n � 75 � x � 47 � 4 � s � 2 � 21� µ �(b) n � 90 � x � 18 � 1 � s � 3 � 88� µ �(c) n � 85 � x � 40 � 8 � s � 3 � 46� µ �(d) n � 110 � x � 58 � 4 � s � 2 � 77� µ �6.(1 pt) Studies have suggusted that twins, in their early

years, tend to have lower IQs and pick up language more slowly

than nontwins. The slower intellectual growth might be causedby benign parental neglect. Suppose it is desired to estimate themean attention time given to twins per week by their parents. Asample of 70 sets of 2 year old boys is taken, and after 1 weekthe attention time recieved was recorded. The data (in hours)calculated the mean at 25 � 8 and the standard deviation at 11 � 9.Use this information to contruct a 95% confidence interval forthe mean attention time given to al twin boys by their parents.� µ �

7.(1 pt)Use the given data to find the 95% confidence interval esti-

ate of the population mean µ. Assume that the population has anormal distribution.

IQ scores of professional athletes:Sample size n � 25Mean x � 104Standard deviation s � 13 � µ �

8.(1 pt) Suppose you have selected a random sample ofn � 10 measurements from a normal distribution. Compare thestandard normal z values with the corresponding t values if youwere forming the following confidence intervals.

(a) 98% confidence intervalz �t �

(b) 95% confidence intervalz �t �

(c) 90% confidence intervalz �t �

9.(1 pt)Weights of 10 red and 36 brown randomly chosen M&M

plain candies are listed below.

Red:0 � 891 0 � 913 0 � 933 0 � 909 0 � 9520 � 924 0 � 898 0 � 907 0 � 882 0 � 877

Brown:

0 � 9 0 � 936 0 � 857 0 � 915 0 � 858 0 � 8970 � 931 0 � 909 0 � 92 0 � 955 0 � 918 0 � 9020 � 867 0 � 86 0 � 877 0 � 931 0 � 904 0 � 920 � 889 0 � 93 0 � 905 0 � 902 0 � 871 0 � 9230 � 985 0 � 921 0 � 929 0 � 988 0 � 986 0 � 8560 � 866 0 � 909 0 � 93 0 � 861 0 � 898 0 � 914

1. To construct a 90% confidence interval for the meanweight of red M&M plain candies, you have to use� A. The t distribution with 11 degrees of freedom� B. The normal distribution� C. The t distribution with 10 degrees of freedom� D. The t distribution with 9 degrees of freedom� E. None of the above

1

2. A 90% confidence interval for the mean weight of redM&M plain candies is � µ �

3. To construct a 90% confidence interval for the meanweight of brown M&M plain candies, you have to use� A. The t distribution with 36 degrees of freedom� B. The t distribution with 35 degrees of freedom� C. The t distribution with 37 degrees of freedom� D. The normal distribution� E. None of the above

4. A 90% confidence interval for the mean weight of brownM&M plain candies is � µ �

10.(1 pt) The following random sample was selected from anormal distribution:

15 12 12 8 19 3 1 1 16 7

(a) Construct a 90% confidence interval for the populationmean µ. � µ �

(b) Construct a 95% confidence interval for the populationmean µ. � µ �

11.(1 pt) The scientific productivity of major world cities wasthe subject of a recent study. The study determined the numberof scientific papers published between 1994 and 1997 by re-searchers from each of the 20 world cities, and is shown below.

City Number of papers City Number of papersCity 1 28 City 11 11City 2 26 City 12 22City 3 29 City 13 27City 4 7 City 14 25City 5 19 City 15 29City 6 6 City 16 5City 7 19 City 17 3City 8 14 City 18 16City 9 17 City 19 29City 10 20 City 20 15

Construct a 80 % confidence interval for the average numberof papers published in major world cities.� µ �

12.(1 pt) The standard IQ test is designed so that the meanis 100 and the standard deviation is 15 for the population of alladults. We wish to find the sample size necessary to estimatethe mean IQ score of statistics students. Suppose we want to be90% confident that our sample mean is within 1 � 5 IQ points ofthe true mean. The mean for this population is clearly greaterthan 100 . The standard deviation for this population is probablyless than 15 because it is a group with less variation than a grouprandomly selected from the general population; therefore, if weuse σ � 15 � we are being conservative by using a value that will

make the sample size at least as large as necessary. Assume thenthat σ � 15 and determine the required sample size.

Answer:

13.(1 pt) Periodically, the county Water Department tests thedrinking water of homeowners for contminants such as lead andcopper. The lead and copper levels in water specimens collectedin 1998 for a sample of 10 residents of a subdevelopement of thecounty are shown below.

lead (µg/L) copper (mg/L)2 � 5 0 � 3792 � 2 0 � 0320 � 1 0 � 7810 0 � 192

0 � 6 0 � 0221 � 5 0 � 7426 0 � 866

5 � 2 0 � 8033 � 1 0 � 4751 � 9 0 � 244

(a) Construct a 99% confidence interval for the mean leadlevel in water specimans of the subdevelopment.� µ �

(b) Construct a 99% confidence interval for the mean copperlevel in water specimans of the subdevelopment.� µ �

14.(1 pt) Suppose that the minimum and maximum ages fortypical textbooks currently used in college courses are 0 and 8years. Use the range rule of thumb to estimate the standard de-viation.

Standard deviation =Find the size of the sample required to estimage the mean age

of textbooks currently used in college courses. Assume that youwant 92% confidence that the sample mean is within 0 � 25 yearof the population mean.

Required sample size =

15.(1 pt) A poll is taken in which 307 out of 600 randomlyselected voters indicated their preference for a certain candidate.Find a 80% confidence interval for p.� p �

16.(1 pt) Use the given confidence interval limits to find thepoint estimate p and the margin of error E �

0 � 74 � p � 0 � 8p � E �17.(1 pt) Astronaunts often report that there are times when

they become disoriented as they move around in zero-gravity.Therefore, they ususally rely on bright colors and other visualinformation to help them estabish a top-down orientation. Astudy was conducted to assses the potential of using color asbody orienting. 90 college students, reclining on their backs inthe dark, found it difficult to establish orientation when posi-tioned on under a rotating disk. This rotating disk was painted

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half black and half white. Out of the 90 students, 73 believedthey were right side up when the white was on top.

Use this information to estimate the true proportion of sub-jects who use the white color as a cue for right-side-up orien-tation. That is, construct a 80% confidence interval for the trueproportion. � p �

18.(1 pt) Construct the 90% confidence interval estimate ofthe population proportion p if the sample size is n � 100 and thenumber of successes in the sample is x � 79 �� p �

19.(1 pt) The EPA wants to test a randomly selected sampleof n water specimens and estimate the mean daily rate of pollu-tion produced by a mining operation. If the EPA wants a 95%confidence interval with a bound of error of 2 milligram per liter(mg/L), how many water specimens are required in the sam-ple? Assume prior knowledge indicates that pollution readingsin water samples taken during a day have been approximatelynormally distributed with a standard deviation of 5 � 2 (mg/L).

n �20.(1 pt) College officials want to estimate the percentage of

students who carry a gun, knife, or other such weapon. Howmany randomly selected students must be surveyed in order tobe 90% confident that the sample percentage has a margin oferror of 1 � 5 percentage points?

(a) Assume that there is no available information that couldbe used as an estimate of p.

Answer:(b) Assume that another study indicated that 7% of college

students carry weapons.

Answer:

21.(1 pt) A random sample of elementary school children inNew York state is to be selected to estimate the proportion pwho have received a medical examination during the past year.An interval estimate of the proportion p with a bound of 0 � 07and 99% confidence is required.

(a) Assuming no prior information about p is available, ap-proximately how large of a sample size is needed?n �

(b) If a planning study indicates that p is around 0 � 3, ap-proximately how large of a sample size is needed?n �

22.(1 pt) Find the critical values χ2L � χ2

1 2 α = 2 and χ2R � χ2

α = 2that correspond to 98% degree of confidence and the sample sizen � 15 �

χ2L � χ2

R �23.(1 pt) According to the Food and Drug Administration

(FDA), a cup of coffee contains on average 115 miligrams (mg)of caffeine, with the amount per cup ranging from 60 to 180 mg.Suppose you want to repeat the FDA experiment to obtain an es-timate of the mean caffeine content in a cup of coffee correct towitin 3 � 9 mg with 90% confidence. How many cups of coffeewould have to be included in your sample?

n �24.(1 pt) Find the minimum sample size needed to be 95%

confident that the sample variance is within 10% of the popula-tion variance.


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1.(1 pt)Type I error is:� A. Deciding alternative hypothesis is true when it is true� B. Deciding alternative hypothesis is true when it is

false� C. Deciding null hypothesis is false when it is true� D. Deciding null hypothesis is true when it is false� E. All of the above� F. None of the above

Type II error is:� A. Deciding null hypothesis is true when it is false� B. Deciding alternative hypothesis is false when it istrue� C. Deciding alternative hypothesis is true when it is true� D. Deciding null hypothesis is false when it is true� E. All of the above� F. None of the above

2.(1 pt) For each statement, express the null hypothesis H0

and alternative hypothesis H1 in symbolic form.1. At least one-half of all Internet users make on-line pur-

chases.� A. H0 : p � 0 � 5 � H1 : p " 0 � 5� B. H0 : µ � 0 � 5 � H1 : µ � 0 � 5� C. H0 : p � 0 � 5 � H1 : p � 0 � 5� D. H0 : µ � 0 � 5 � H1 : µ " 0 � 52. IQ scores of statistics students have a standard deviation

less than 15.� A. H0 : µ � 15 � H1 : µ " 15� B. H0 : σ � 15 � H1 : σ � 15� C. H0 : µ � 15 � H1 : µ � 15� D. H0 : σ � 15 � H1 : σ " 15

3. The mean salary of statistics professors is less than70,000 dollars.� A. H0 : µ � 70 � 000 � H1 : µ " 70 � 000� B. H0 : µ " 70 � 000 � H1 : µ � 70 � 000� C. H0 : µ � 70 � 000 � H1 : µ � 70 � 000� D. H0 : µ � 70 � 000 � H1 : µ � 70 � 000

3.(1 pt) Given the significance level α � 0 � 095 find the fol-lowing:

(a) lower-tailed z valuez �

(b) right-tailed z valuez �

(c) two-tailed z value� z �>�4.(1 pt) Find the critical z value for a left-tailed test using a

significance level of α � 0 � 01 �

5.(1 pt) Find the critical z value using a significance level ofα � 0 � 1 if the alternative hypothesis H0 is µ " 26.

6.(1 pt) A random sample of 100 observations from a popu-lation with standard deviation 33 � 612 yielded a sample mean of96.(a) Given that the null hypothesis is µ � 90 and the alternativehypothesis is µ " 90 using α �1� 05, find the following:

(i) critical z score(ii) test statistic �

(b) Given that the null hypothesis is µ � 90 and the alternativehypothesis is µ ?� 90 using α �1� 05, find the following:

(i) the positive critical z score(ii) the negative critical z score(iii) test statistic �The conclusion from part (a) is:� A. Reject the null hypothesis� B. There is insufficient evidence to reject the null hy-

pothesis� C. None of the above

The conclusion from part (b) is:� A. There is insufficient evidence to reject the null hy-pothesis� B. Reject the null hypothesis� C. None of the above

7.(1 pt) It is necessary for an automobile producer to esti-mate the number of miles per gallon achieved by its cars. Sup-pose that the sample mean for a random sample of 150 cars is29 � 9 miles and assume the standard deviation is 2 � 3 miles. Nowsuppose the car producer wants to test the hypothesis that µ, themean number of miles per gallon, is 31 � 5 against the alternativehypothesis that it is not 31 � 5. Conduct a test using α �@� 05 bygiving the following:

(a) positive critical z score(b) negative critical z score(c) test statisticThe final conclustion is� A. There is not sufficient evidence to reject the null hy-

pothesis that µ � 31 � 5.� B. We can reject the null hypothesis that µ � 31 � 5 andaccept that µ ?� 31 � 5.

8.(1 pt) The contents of 32 cans of Coke have a mean ofx � 12 � 15 and a standard deviation of s � 0 � 09 � Find the valueof the test statistic z for the claim that the population mean isµ � 12 �

9.(1 pt) Golf-course designers have become concerned thatold courses are becoming obsolete since new technology hasgiven golfers the ability to hit the ball so far. Designers, there-fore, have proposed that new golf courses need to be built so that

1

an average golfer can be expected to hit the ball more than 245yards on average. Suppose a random sample of 190 golfers bechosen so that their mean driving distance is 245 � 3 yards, witha standard deviation of 41 � 9.

Conduct a hypothesis test where H0 : µ � 245 andH1 : µ " 245 by computing the following:(a) test statistic(b) p-value p �(c) If this was a two-tailed test, then the p-value is

10.(1 pt) Assume you are using a significance level of α �0 � 05 to test the claim that µ � 16 and that your sample is a ran-dom sample of 38 values. Find β, the probability of making atype II error (failing to reject a false null hypothesis), given thatthe population actually has a normal distribution with µ � 10and σ � 8 �

β �11.(1 pt) Physicians at a clinic gave what they thought were

drugs to 970 asthma, ulcer, and herpes patients. Although thedoctors later learned that the drugs were really placebos, 52 %of the patients reported an improved condition. Assume that ifthe placebo is ineffective, the probability of a patients condi-tion improving is 0 � 5. For the hypotheses that the proportion ofimproving is 0 � 5 against that it is " 0 � 5, find the p-value.

p �12.(1 pt) Test the claim that the population of sophomore col-

lege students has a mean grade point average greater than 2 � 1.Sample statistics include n � 110, x � 2 � 4, and s � 0 � 9. Use asignificance level of α � 0 � 03.

The test statistic isThe critical value isThe P-Value isThe final conclustion is� A. There is sufficient evidence to support the claim that

the mean grade point average is greater than 2.1.� B. There is not sufficient evidence to support the claimthat the mean grade point average is greater than 2.1.

13.(1 pt) 50 people are randomly selected and the accuracyof their wristwatches is checked, with positive errors represent-ing watches that are ahead of the correct time and negative er-rors representing watches that are behind the correct time. The50 values have a mean of 114sec and a standard deviation of246sec. Use a 0 � 01 significance level to test the claim that thepopulation of all watches has a mean of 0sec.

The test statistic isThe P-Value isThe final conclustion is� A. There is sufficient evidence to warrant rejection of

the claim that the mean is equal to 0� B. There is not sufficient evidence to warrant rejectionof the claim that the mean is equal to 0

14.(1 pt) A sample of 10 measurments, randomly selectedfrom a normally distributed population, resulted in a sample

mean, x � 9 � 6 and sample standard deviation s � 1 � 23. Usingα � 0 � 01, test the null hypothesis that the mean of the popula-tion is 8 � 1 against the alternative hypothesis that the mean of thepopulation, µ � 8 � 1 by giving the following:(a) the degree of freedom(b) the critical t value(c) the test statistic

The final conclustion is� A. There is not sufficient evidence to reject the null hy-pothesis that µ � 8 � 1.� B. We can reject the null hypothesis that µ � 8 � 1 andaccept that µ � 8 � 1.

15.(1 pt) The effectiveness of a new bug repellent is testedon 13 subjects for a 10 hour period. Based on the number andlocation of the bug bites, the percentage of surface area exposedprotected from bites was calculated for each of the subjects. Theresults were as follows:

x � 93 %, s � 10%The new repellent is considered effective if it provides a per-

cent repellency of at least 99. Using α � 0 � 01, construct a hy-pothesis test with null hypothesis µ � 0 � 99 and alternative hy-pothesis µ " 0 � 99 to determine whether the mean repellency ofthe new bug relellent is greater than 99 by computing the fol-lowing:

(a) the degree of freedom(b) the critical t value(c) the test statisticsThe final conclustion is� A. There is not sufficient evidence to reject the null hy-

pothesis that µ � 0 � 99.� B. We can reject the null hypothesis that µ � 0 � 99 andaccept that µ " 0 � 99, that is, the bug repellent is effec-tive.

16.(1 pt) Test the claim that for the population of statisticsfinal exams, the mean score is µ � 80 � Sample statistics includen � 13 � x � 81 � and s � 12 � Use a significance level of α � 0 � 05 �

The test statistic isThe positive critical value isThe negative critical value isThe conclustion is� A. There is sufficient evidence to warrant rejection of

the claim that the mean score is equal to 80� B. There is not sufficient evidence to warrant rejectionof the claim that the mean score is equal to 80

17.(1 pt) When a poultry farmer uses his regular feed, thenewborn checkens have normally distributed weights with amean of 61 � 6 oz. In an experiment with an enriched feedmixture, ten chickens are born with the following weights (inounces).

64 � 7 � 66 � 9 � 65 � 6 � 67 � 9 � 63 � 4 � 66 � 3 � 65 � 6 � 62 � 1 � 64 � 4 � 66 � 12

Use the α � 0 � 05 significance level to test the claim that themean weight is higher with the enriched feed.

The sample mean is x �The sample standard deviation is s �The test statistic is t �The critical value is t �The conclusion is� A. There is not sufficient evidence to support the claim

that with the enriched feed, the mean weight is greaterthan 61.6.� B. There is sufficient evidence to support the claim thatwith the enriched feed, the mean weight is greater than61.6.

18.(1 pt) One of the most feared predators in the ocean is thegreat white shark. It is known that the white shark grows to amean length of 18 feet; however, one marine biologist believesthat great white sharks off the Bermuda coast grow much longer.To test this claim, full-grown white sharks were captured, mea-sured, and then set free. However, this was a difficult, costly andvery dangerous task, so only four sharks were actually sampled.Their lengths were 26 � 22 � 23 � and 25 feet. Do the data providesufficient evidence to support the claim? Use α � 0 � 05

test statistic t �rejection region t "The final conclustion is� A. We can reject the null hypothesis that the average

length of the shark is 18, and accept that the averagelength of the shark is greater than 18.� B. There is not sufficient evidence to reject the null hy-pothesis that the average length of the shark is 18.

19.(1 pt) A random sample of 110 observations is selectedfrom a binomial population with unknown probability of suc-cess p. The computed value of p is 0 � 63.(1) Test H0 : p � 0 � 65 against H1 : p " 0 � 65. Use α � 0 � 01.

test statistic z �critical z score

The final conclustion is� A. There is not sufficient evidence to reject the null hy-pothesis that p � 0 � 65.� B. We can reject the null hypothesis that p � 0 � 65 andaccept that p " 0 � 65.

(2) Test H0 : p � 0 � 65 against H1 : p � 0 � 65. Use α � 0 � 01.test statistic z �critical z score

The final conclustion is� A. We can reject the null hypothesis that p � 0 � 65 andaccept that p � 0 � 65.� B. There is not sufficient evidence to reject the null hy-pothesis that p � 0 � 65.

(3) Test H0 : p � 0 � 5 against H1 : p ?� 0 � 5. Use α � 0 � 05.test statistic z �positive critical z score

negative critical z scoreThe final conclustion is� A. There is not sufficient evidence to reject the null hy-

pothesis that p � 0 � 5.� B. We can reject the null hypothesis that p � 0 � 5 andaccept that p ?� 0 � 5.

20.(1 pt) According to a recent marketing campaign, 90drinkers of either Diet Coke or Diet Pepsi participated in a blindtaste test to see which of the drinks was their favorite. In onePepsi television commercial, an anouncer states that ”in recentblind taste tests, more than one half of the surveyed preferredDiet Pepsi over Diet Coke.” Suppose that out of those 90, 60preferred Diet Pepsi. Test the hypothesis, using α � 0 � 01 thatmore than half of all participants will select Diet Pepsi in a blindtaste test by giving the following:(a) the test statistic(b) the critical z score

The final conclustion is� A. We can reject the null hypothesis that p � 0 � 5 andaccept that p " 0 � 5.� B. There is not sufficient evidence to reject the null hy-pothesis that p � 0 � 5.

21.(1 pt) A survey of 1650 people who took trips revealedthat 156 of them included a visit to a theme park. Based onthose survery results, a management consultant claims that lessthan 11 % of trips include a theme park visit. Test this claimusing the α � 0 � 01 significance level.

The test statistic isThe critical value isThe conclusion is� A. There is sufficient evidence to support the claim that

less than 11 % of trips include a theme park visit.� B. There is not sufficient evidence to support the claimthat less than 11 % of trips include a theme park visit.

22.(1 pt) A new cream that advertises that it can reduce wrin-kles and improve skin was subject to a recent study. A sample of42 women over the age of 50 used the new cream for 6 months.Of those 42 women, 31 of them reported skin improvement(asjudged by a dermatologist). Is this evidence that the cream willimprove the skin of more than 40% of women over the age of50? Test using α � 0 � 01.

test statistics z �rejection region z "The final conclustion is� A. There is not sufficient evidence to reject the null hy-

pothesis that p � 0 � 4. That is, there is not sufficientevidence to reject that the cream can improve the skinof more than 40% of women over 50.� B. We can reject the null hypothesis that p � 0 � 4 andaccept that p " 0 � 4. That is, the cream can improve theskin of more than 40% of women over 50.

3

23.(1 pt) A random sample of n � 10 observations from anormal population produced the following measurements:

3 3 7 0 6 2 34 7 4

Do the data provide sufficient evidence to indicate that σ2 � 2?Use α � 0 � 01, and compute the following:(a) sample standard deviation s �(b) test statistic χ2 �(c) critical χ2

α �The final conclustion is� A. There is not sufficient evidence to reject the null hy-

pothesis that σ2 � 2.� B. We can reject the null hypothesis that σ2 � 2 andaccept that σ2 � 2.

24.(1 pt) Use a α � 0 � 01 significance level to test the claimthat σ � 13 if the sample statistics include n � 25 � x � 104 � ands � 19 �

The test statistic isThe smaller critical number isThe bigger critical number isWhat is your conclusion?� A. There is sufficient evidence to warrant the rejection

of the claim that the population standard deviation isequal to 13� B. There is not sufficient evidence to warrant the rejec-tion of the claim that the population standard deviationis equal to 13


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1.(1 pt) In order to compare the means of two populations,independent random samples of 438 observations are selectedfrom each population, with the following results:

Sample 1 Sample 2x1 � 5226 x2 � 5468s1 � 100 s2 � 120

(a) Use a 97 % confidence interval to estimate the differencebetween the population means

�µ1 � µ2 � .� �

µ1 � µ2 ��(b) Test the null hypothesis: H0 :

�µ1 � µ2 �� 0 versus the al-

ternative hypothesis: Ha :�µ1 � µ2 �A?� 0. Using α � 0 � 03, give

the following:(i) the test statistic z �(ii) the positive critical z score(iii) the negative critical z scoreThe final conclustion is� A. We can reject the null hypothesis that

�µ1 � µ2 �� 0

and accept that�µ1 � µ2 ��?� 0.� B. There is not sufficient evidence to reject the null hy-

pothesis that�µ1 � µ2 �� 0.

(c) Test the null hypothesis: H0 :�µ1 � µ2 �� 28 versus the al-

ternative hypothesis: Ha :�µ1 � µ2 ��?� 28. Using α � 0 � 03, give

the following:(i) the test statistic z �(ii) the positive critical z score(iii) the negative critical z scoreThe final conclustion is� A. We can reject the null hypothesis that

�µ1 � µ2 �� 28

and accept that�µ1 � µ2 ��?� 28.� B. There is not sufficient evidence to reject the null hy-

pothesis that�µ1 � µ2 �� 28.

2.(1 pt) Test the claim that the two samples described belowcome from populations with the same mean. Assume that thesamples are independent simple random samples. Use a signifi-cance level of 0 � 05.Sample 1: n1 � 82 � x1 � 10 � s1 � 1.Sample 2: n2 � 54 � x2 � 15 � s2 � 3.

The test statistic isThe P-Value isThe conclusion is� A. There is not sufficient evidence to warrant rejection

of the claim that the two populations have the samemean.� B. There is sufficient evidence to warrant rejection ofthe claim that the two populations have the same mean.

3.(1 pt) The purpose of this question is to compare the vari-ability of x1 and x2 with the variability of

�x1 � x2 � .

(a) Suppose the first sample of 100 observations is selectedfrom a population with mean µ1 � 190 and variance σ2

1 � 1510.Construct an interval extending 2 standard deviations of x1 oneach side of µ1. � µ1 �

(b) Suppose the second sample of 100 observations is se-lected from a population with mean µ2 � 190 and varianceσ2

2 � 870. Construct an interval extending 2 standard deviationsof x2 on each side of µ2. � µ2 �

(c) Consider the difference between the two sample means�x1 � x2 � . Compute the mean and the standard deviation of the

sampling distribution of�x1 � x2 � .

mean =standard deviation =

(d) Based on 100 observations, construct an interval extend-ing 2 standard deviations of

�x1 � x2 � on each side of

�µ1 � µ2 ��

µ1 � µ2 ��4.(1 pt) Randomly selected 60 student cars have ages with a

mean of 7 � 5 years and a standard deviation of 3 � 4 years, whilerandomly selected 55 faculty cars have ages with a mean of 5 � 8years and a standard deviation of 3 � 7 years.

1. Use a 0 � 01 significance level to test the claim that studentcars are older than faculty cars.

The test statistic isThe critical value isIs there sufficient evidence to support the claim that student

cars are older than faculty cars?� A. No� B. Yes

2. Construct a 99% confidence interval estimate of the dif-ference µ1 � µ2, where µ1 is the mean age of student cars and µ2

is the mean age of faculty cars.� �µ1 � µ2 ��

5.(1 pt) Two independent samples have been selected, 83 ob-servations from population 1 and 72 observations from popula-tion 2. The sample means have been calculated to be x1 � 5 andx2 � 7. From previous experience with these populations, it isknown that the variances are σ2

1 � 31 and σ22 � 36.

(a) Find σ B x1 2 x2 C .answer:

(b) Determine the rejection region for the test of H0 :�µ1 � µ2 �� 2 � 28 and Ha :

�µ1 � µ2 �D" 2 � 28 Use α � 0 � 03.

z "(c) Compute the test statistic.

z �The final conclustion is� A. There is not sufficient evidence to reject the null hy-

pothesis that�µ1 � µ2 �� 2 � 28.

1

� B. We can reject the null hypothesis that�µ1 � µ2 �E�

2 � 28 and accept that�µ1 � µ2 �D" 2 � 28.

(d) Construct a 97 % confidence interval for�µ1 � µ2 � .� �

µ1 � µ2 ��6.(1 pt) Test the claim that the two samples described below

come from populations with the same mean. Assume that thesamples are independent simple random samples. Use a signif-cance level of α � 0 � 01Sample 1: n1 � 3 � x1 � 29 � 1 � s1 � 5 � 48Sample 2: n2 � 5 � x2 � 22 � s2 � 3 � 57(a) The degree of freedom is(b) The test statistic is(c) Determine the rejection region for the test of H0 :

�µ1 �

µ2 �� 0 and Ha :�µ1 � µ2 ��?� 0� t �F"

The final conclustion is� A. There is not sufficient evidence to reject the null hy-pothesis that

�µ1 � µ2 �� 0.� B. We can reject the null hypothesis that

�µ1 � µ2 �� 0

and accept that�µ1 � µ2 ��?� 0.

7.(1 pt) Test the given claim using the α � 0 � 05 significancelevel and assuming that the populations are normally distributed.

Claim: The treatment population and the placebo populationhave the same mean.

Treatment group: n � 6 � x � 82 � s � 6 � 9 �Placebo group: n � 12 � x � 74 � s � 6 � 7 �The test statistic isThe positive critical value isThe negative critical value isIs there sufficient evidence to warrant the rejection of the

claim that the treatment and placebo populations have the samemean?� A. No� B. Yes

8.(1 pt) Randomly selected students were given five secondsto estimate the value of a product of numbers with the resultsshown below.

Estimates from students given 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8:

10000 � 6000 � 2040 � 45000 � 842 � 40320 � 42200 � 40320 � 1560 � 10000

Estimates from students given 8 G 7 G 6 G 5 G 4 G 3 G 2 G 1:

40000 � 550 � 225 � 3600 � 100000 � 377 � 400 � 23410 � 100000 � 3000

Use a 0 � 05 significance level to test the following claims:1. Claim: the two populations have equal variances.The test statistic isThe larger critical value isThe conclusion is� A. There is sufficient evidence to warrant the rejection

of the claim that the two populations have equal vari-ances

� B. There is not sufficient evidence to warrant the rejec-tion of the claim that the two populations have equalvariances

2. Claim: the two populations have the same mean.The test statistic isThe positive critical value isThe negative critical value isThe conclusion is� A. There is sufficient evidence to warrant the rejection

of the claim that the two populations have the samemean� B. There is not sufficient evidence to warrant the rejec-tion of the claim that the two populations have the samemean

9.(1 pt) Suppose you want to test the claim the the pairedsample data given below come from a population for which themean difference is µd � 0.

x 62 76 55 60 64 66 67y 80 64 63 64 67 59 83

Use a 0 � 05 significance level to find the following:(a) The mean value of the differnces d for the paired sample

datad �

(b) The standard deviation of the differences d for the pairedsample datasd �

(c) The t test statistict �

(d) The positive critical valuet �

(e) The negative critical valuet �

(f) Does the test statistic fall in the critical region?� A. Yes� B. No

(g) Construct a 95% conficence interval for the populationmean of all differences x � y. � µd �

10.(1 pt) Ten randomly selected people took an IQ test A,and next day they took a very similar IQ test B. Their scores areshown in the table below.

Person A B C D E F G H I JTest A 108 95 91 118 125 78 82 103 96 107Test B 108 94 90 118 124 79 84 107 96 111

1. Use a 0 � 05 significance level to test the claim that peopledo better on the second test than they do on the first.

The test statistic isThe critical vaue isIs there sufficient evidence to support the claim that people

do better on the second test?2

� A. Yes� B. No

2. Construct a 95% confidence interval for the mean of thedifferences. � µ �

11.(1 pt) A paired difference experiment yielded nD pairs ofobservations. In each case described below, what is the rejectionregion for testing H0 : µ � 6 against Ha : µ " 6? Use sD � 12 � 1.(a) nD � 48 � α � 0 � 04

z "(b) nD � 5 � α � 0 � 08

t "(c) nD � 17 � α � 0 � 05

t "12.(1 pt) A paired difference experiment produced the fol-

lowing results:

nD � 41 � x1 � 187 � x2 � 184 � xD � 3 � sD � 70 �(a) Determine the rejection region for the hypothesis H0 : µD �0 if Ha : µD " 0. Use α � 0 � 09.

z "(b) Conduct a paired difference test described above.

The test statistic isThe final conclustion is� A. There is not sufficient evidence to reject the null hy-

pothesis that µD � 0.� B. We can reject the null hypothesis that µD � 0 andaccept that µD " 0.

13.(1 pt) In a study of red/green color blindness, 600 menand 2050 women are randomly selected and tested. Among themen, 52 have red/green color blindness. Among the women, 5have red/green color blindness. Test the claim that men have ahigher rate of red/green color blindness.

The test statistic isIs there sufficient evidence to support the claim that men have

a higher rate of red/green color blindness than women?� A. Yes� B. No

Construct the 96% confidence interval for the difference be-tween the color blindness rates of men and women.� �

p1 � p2 ��14.(1 pt) Independent random samples, each containing 700

observations, were selected from two binomial populations. Thesamples from populations 1 and 2 produced 253 and 358 suc-cesses, respectively.(a) Test H0 :

�p1 � p2 �E� 0 against Ha :

�p1 � p2 �H?� 0. Use

α � 0 � 04test statistic �rejection region � z �I"The final conclustion is� A. We can reject the null hypothesis that

�p1 � p2 �� 0

and accept that�p1 � p2 ��?� 0.

� B. There is not sufficient evidence to reject the null hy-pothesis that

�p1 � p2 �� 0.

(b) Test H0 :�p1 � p2 �D� 0 against Ha :

�p1 � p2 ��" 0. Use

α � 0 � 06test statistic �rejection region z "The final conclustion is� A. There is not sufficient evidence to reject the null hy-

pothesis that�p1 � p2 �� 0.� B. We can reject the null hypothesis that

�p1 � p2 �� 0

and accept that�p1 � p2 ��" 0.

15.(1 pt) Suppose a group of 800 smokers (who all wanted togive up smoking) were randomly assigned to recieve an antide-pressant drug or a placebo for six weeks. Of the 525 patientswho recieved the antidepressant drug, 10 were not smoking oneyear later. Of the 275 patients who recieved the placebo, 75were not smoking one year later. Given the null hypothesis H0 :�p1 � p2 �� 0 and the alternative hypothesis Ha :

�p1 � p2 ��?� 0,

conduct a test to see if taking an antidepressant drug can helpsmokers stop smoking. Use α � 0 � 08(a) The rejection region is � z �I"(b) The test statistic is z �The final conclustion is� A. We can reject the null hypothesis that

�p1 � p2 �� 0

and accept that�p1 � p2 ��?� 0.� B. There is not sufficient evidence to reject the null hy-

pothesis that�p1 � p2 �� 0.

16.(1 pt) Test the given claim using the α � 0 � 02 significancelevel and assuming that the populations are normally distributed.

Claim: The treatment population and the placebo populationhave different variances.

Treatment group: n � 8 � x � 73 � 9 � s � 15 � 9 �Placebo group: n � 5 � x � 60 � 1 � s � 11 � 9 �The test statistic isThe larger critical value isWhat is your conclusion?� A. There is sufficient evidence to support the claim that

the treatment and placebo populations have differentvariances.� B. There is not sufficient evidence to support the claimthat the treatment and placebo populations have differ-ent variances.

17.(1 pt) Suppose you wanted to estimate the difference be-tween two population means correct to within 4 with probability0 � 95. If prior information suggests that the popluation variancesare approximately equal to σ2

1 � σ22 � 11 and you want to select

independent random samples of equal size from the poulations,how large should the sample sizes, n1 and n2 be?answer: n1 � n2 �

18.(1 pt) Find the size of each sample needed to estimate thedifference between the proportions of boys and girls under 10

3

years old who are afraid of spiders. Assume that we want 99%confidence that the error is smaller than 0 � 07 �

n �19.(1 pt)The sample size needed to estimate the difference between

two population proportions to within a margin of error E with asignificance level of α can be found as follows. In the expres-sion

E � zα = 2 J p1q1

n1

p2q2

n2

we replace both n1 and n2 by n (assuming that both sampleshave the same size) and replace each of p1 � p2 � q1 � and q2 by

0 � 5 (because their values are not known). Then we solve for n,and get

n � �zα = 2 � 22E2 �

Finally, increase the value of n to the next larger integer num-ber.

Use the above formula to find the size of each sample neededto estimate the difference between the proportions of boys andgirls under 10 years old who are afraid of spiders. Assume thatwe want 99% confidence that the error is smaller than 0 � 09 �

n �


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1.(1 pt)

Use a scatterplot and the linear correlation coefficient r to determine whether there is a correlation between the two variables.

x 0 � 3 1 � 3 2 � 2 3 4 � 5 5 � 5 6 � 3 7 � 1 8 � 9 9 � 9 10 11 � 5 12 � 3 13 � 8 14 � 8y 14 � 7 13 � 7 12 � 8 12 10 � 5 9 � 5 8 � 7 7 � 9 6 � 1 5 � 1 5 3 � 5 2 � 7 1 � 2 0 � 2

r �There is� A. a positive correlation between x and y� B. a nonlinear correlation between x and y� C. a negative correlation between x and y� D. a perfect positive correlation between x and y� E. a perfect negative correlation between x and y� F. no correlation between x and y

2.(1 pt) Match the following sample correlation coeffi-cients with the explaination of what that correlation coeffiecientmeans.

1. r �K� 12. r �K� 13. r � 04. r �K� 92

A. a weak positive relationship between x and yB. a perfect negative relationship between x and yC. a strong positive relationship between x and yD. no relationship between x andy

3.(1 pt) Given the following data set,

x 4 2 4 0 4 � 2 4y 5 3 1 4 6 0 6

Compute the coefficient of correlation rr �

4.(1 pt) Heights (in centimeters) and weights (in kilograms)of 7 supermodels are given below. Find the regression equation,letting the first variable be the independent

�x � variable, and pre-

dict the weight of a supermodel who is 173 cm tall.

Height 176 176 174 176 172 166 178Weight 56 55 54 54 52 47 57

The regression equation is y � x �The best predicted weight of a supermodel who is 173 cm

tall is .

5.(1 pt) Is the number of games won by a major league base-ball team in a season related to the team batting average? Thetable below shows the number of games won and the battingaverage of 8 teams.

Team Games Won Batting Average1 78 0 � 2762 82 0 � 2773 89 0 � 264 88 0 � 2815 93 0 � 2666 88 0 � 2867 94 0 � 2658 103 0 � 274

Using games won as the independent variable x, do the follow-ing:(a) The correlation coefficient is

r �(b) The equation of the least squares line is

y � x

6.(1 pt) The amounts of 6 restaurant bills and the correspond-ing amounts of the tips are given in the below.

Bill 52 � 44 32 � 98 43 � 58 64 � 30 97 � 34 106 � 27Tip 7 � 00 4 � 50 5 � 50 7 � 70 16 � 00 16 � 00

Use a 0.05 confidence level to find the following:The test statistic r �Is there a significant correlation?� A. Yes� B. No

The regression equation is y � x �If the amount of the bill is $65 � the best prediction for the

amount of the tip is

7.(1 pt) The amounts of 6 restaurant bills and the correspond-ing amounts of the tips are given in the below.

Bill 88 � 01 49 � 72 70 � 29 43 � 58 52 � 44 97 � 34Tip 10 � 00 5 � 28 10 � 00 5 � 50 7 � 00 16 � 00

Use a 0.05 confidence level to find the following:1

The test statistic r �The test statistic t �The critical value t �Is there a significant correlation?� A. Yes� B. No

The regression equation is y � x �If the amount of the bill is $60 � the best prediction for the

amount of the tip is ,and a prediction interval estimate of the amount amount of thetip is � tip �

8.(1 pt) Construct both a 80% and a 95% confidence intervalfor β1.β1 � 45 � s � 7 � 1 � SSxx � 69 � n � 2380% : � β1 �95% : � β1 �

9.(1 pt) Find the multiple regression equation for the datagiven below.

x1 � 2 � 2 � 1 2 3x2 � 3 1 � 1 � 2 2y 1 � 12 � 1 19 12

The equation is y � x1 x2 �

10.(1 pt) Consider the data set below.

x 3 5 3 3 4 8y 2 7 9 8 5 3

For a hyopthesis test, where H0 : β1 � 0 and H1 : β1 ?� 0, andusing α � 0 � 05, give the following:(a) The test statistic

t �(b) The degree of freedom

d f �(c) The rejection region� t �F"The final conclustion is� A. There is not sufficient evidence to reject the null hy-

pothesis that β1 � 0.� B. We can reject the null hypothesis that β1 � 0 andaccept that β1 ?� 0.

11.(1 pt) In some cases, the best-fitting multiple regressionequation is of the form y � b0

b1x b2x2 b3x3 � The graphof such an equation is called a cubic. Using the data set given

below, and letting x1 � x � x2 � x2 � and x3 � x3 � find the multipleregression equation for the cubic that best fits the given data.

x � 10 � 6 � 4 0 4 5 8y 40 � 2 9 � 3 6 � 4 8 � 5 7 � 5 4 � 8 � 15 � 1

The equation is y � x x2 x3 �12.(1 pt) A study was conducted to detemine whether a the

final grade of a student in an introductory psychology courseis linearly related to his or her performance on the verbal abilitytest administered before college entrance. The verbal scores andfinal grades for 10 students are shown in the table below.

Student Verbal Score x Final Grade y1 26 672 56 683 60 864 30 755 62 836 80 877 74 948 31 949 62 74

10 40 98

Find the following:(a) The correlation coefficient

r �(b) The least squares line

y � x13.(1 pt) For each paired data set, construct a scatterplot and

identify the mathematical model that best fits the given data.x 1 2 3 4 5 6 7y 4 � 5 7 � 5 10 � 5 13 � 5 16 � 5 19 � 5 22 � 5� A. Power� B. Linear� C. Logistic� D. Quadratic� E. Exponentialx 1 2 3 4 5 6 7y 5 � 5 2 � 5 1 � 5 2 � 5 5 � 5 10 � 5 17 � 5� A. Exponential� B. Quadratic� C. Linear� D. Power� E. Logarithmic


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1.(1 pt) A multinomial experiment with k � 3 cells andn � 280 produced the data shown below.

Cell 1 Cell 2 Cell 3ni 68 79 133

If the null hypothesis is H0 : p1 �K� 25 � p2 �L� 25 � p3 �K� 5 andusing α � 0 � 01, then do the following:(a) Find the expected value of Cell 1.

E(Cell 1) �(b) Find the expected value of Cell 2.

E(Cell 2) �(c) Find the expected value of Cell 3.

E(Cell 3) �(d) Find the test statistic.

χ2 �(e) Find the rejection region.

χ2 "The final conclustion is� A. There is not sufficient evidence to reject the null hy-

pothesis that p1 �1� 25 � p2 �1� 25 � p3 �K� 5.� B. We can reject the null hypothesis that p1 �#� 25 � p2 �� 25 � p3 �@� 5 and accept that at least one of the multi-nomial probabilities does not equal its hypothesizedvalue.

2.(1 pt) A computer random number generator was used togenerate 650 random digits (0,1,...,9). The observed frequencesof the digits are given in the table below.

0 1 2 3 4 5 6 7 8 964 64 56 54 56 57 56 58 58 127

Test the claim that all the outcomes are equally likely usingthe significance level α � 0 � 01.

The expected frequency of each outcome is E �The test statistic is χ2 �The critical value is χ2 �Is there sufficient evidence to warrant the rejection of the

claim that all the outcomes are equally likely?� A. Yes� B. No3.(1 pt) It has been suggusted that the highest priority of re-

tirees is travel. Thus, a study was conducted to investigate thedifferences in the length of stay of a trip for pre and postretirees.A sample of 704 travelers were asked how long they stayed on atypical trip. The observed results of the study are found below.

Number of Nights Pre-retirement Post-retirement Total4 � 7 248 170 418

8 � 13 81 70 15114 � 21 31 52 83

22 or more 13 39 52Total 373 331 704

With this information, construct a table of estimated expectedvalues.

Number of Nights Pre-retirement Post-retirement4 � 78 � 13

14 � 2122 or more

Now, with that information, determine whether the length ofstay is independent of retirement using α � 0 � 01

χ2 �rejection region is χ2 "The final conclustion is� A. There is not sufficient evidence to reject the null hy-

pothesis that the length of stay is independent of retire-ment.� B. We can reject the null hypothesis that the length ofstay is independent of retirement and accept the alter-native hypothesis that the two are dependent.

4.(1 pt) Among drivers who have had a car crash in the lastyear, 160 were randomly selected and categorized by age, withthe results listed in the table below.

Age Under 25 25-44 45-64 Over 64Drivers 63 38 21 38

If all ages have the same crash rate, we would expect (be-cause of the age distribution of licensed drivers) the given cat-egories to have 16%, 44%, 27%, 13% of the subjects, respec-tively. At the 0.025 significance level, test the claim that thedistribution of crashes conforms to the distribution of ages.

The test statistic is χ2 �The critical value is χ2 �The conclusion is� A. There is sufficient evidence to warrant the rejection

of the claim that the distribution of crashes conforms tothe distibuion of ages.� B. There is not sufficient evidence to warrant the rejec-tion of the claim that the distribution of crashes con-forms to the distibuion of ages.

5.(1 pt) Test the null hypothesis of independence of the twoclassifications, A and B, of the 3 G 3 contingency table shownbelow. Test using α � 0 � 05

B1 B2 B3 TotalA1 53 49 56 158A2 71 69 73 213A3 55 41 54 150

Total 179 159 183 5211

χ2 �rejection region is χ2 "The final conclustion is� A. There is not sufficient evidence to reject the null hy-

pothesis that A and B are independent.� B. We can reject the null hypothesis that A and B areindependent and accept that A and B are dependent.

6.(1 pt) The number of men and women among professors inMath, Physics, Chemistry, Linguistics, and English departmentsof a certain college were counted, and the results are shown inthe table below.

Dept. Math Physics Chemistry Linguistics EnglishMen 53 71 33 15 33

Women 2 4 4 2 19

Test the claim that the gender of a professor is independentof the department. Use the significance level α � 0 � 025

The test statistic is χ2 �The critical value is χ2 �Is there sufficient evidence to warrant the rejection of the

claim that the gender of a professor is independent of the de-partment?� A. Yes� B. No


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1.(1 pt) Complete the ANOVA table for a completely ran-domized design below.

Source df SS MS FTreatments 12 15 � 1

ErrorTotal 48 42 � 8

2.(1 pt) The table below lists the body temperatures of sixrandomly selected subjects from each of three different agegroups. Use the α � 0 � 05 significance level to test the claimthat the three age-group populations have different mean bodytemperatures.

16-20 21-29 30 and oldersubject 1 98.2 98.6 97.4subject 2 97.6 98.3 97.1subject 3 98.4 98.8 97.5subject 4 97.3 98.8 97subject 5 98.8 97.6 97.6subject 6 98.1 97.9 98.2

mean 98.067 98.333 97.467standard deviation 0.543 0.497 0.427

The variance between samples is ns2x �

The variance within samples is s2p �

The test statistic is F �The critical value is F �Is there sufficient evidence to warrant the rejection of the

claim that the three age-group populations have the same meanbody temperature?� A. No� B. Yes

3.(1 pt) An experiment is conducted to determine whetherthere is a differnce among the mean increases in growth pro-duced by five inculins (A, B, C, D and E) of growth hormonesfor plants. The experimental material consists of 20 cuttings ofa shrub (all of equal weight), with four cuttings randomly as-signed to each of the five different inoculins. The increase inweight and the standard deviation of the experiment are given inthe table below.

A B C D EPlant 1 18 27 21 12 9Plant 2 12 15 23 12 12Plant 3 19 20 23 15 14Plant 4 16 20 23 14 7Mean 16 � 25 20 � 5 22 � 5 13 � 25 10 � 5

Standard Dev. 2 � 6810 4 � 2720 0 � 8660 1 � 2990 2 � 6926

Compute the following:(a) SST �

(b) SSE �(c) MST �(d) MSE �(e) F �

4.(1 pt)Which of the following changes the analysis of variance results?� A. each of the sample values is multiplied by the same

constant� B. each value in one of the samples is multiplied by thesame constant� C. each of the sample values is converted to a differentscale� D. the order of the samples is changed� E. the same constant is added to each value in one ofthe samples� F. the same constant is added to every one of the samplevalues

5.(1 pt) 176 264 Suppose the Total Sum of Squares for a com-pletely randomzied design with p � 5 treatments and n � 15 to-tal measurements (SS(Total))is equal to 440. In each of the fol-lowing cases, conduct an F -test of the null hypothesis that themean responses for the 5 treatments are the same. Use α � 0 � 01.(a) Sum of Squares for Treatment (SST) is 40% of SS(Total)

F �Rejection region F "

The final conclustion is� A. There is not sufficient evidence to reject the null hy-pothesis that the mean responses for the treatments arethe same.� B. We can reject the null hypothesis that the mean re-sponses for the treatments are the same and accept thealternative hypothesis that at least two treatment meansdiffer.

(b) Sum of Squares for Treatment (SST) is 80% of SS(Total)F �Rejection region F "

The final conclustion is� A. There is not sufficient evidence to reject the null hy-pothesis that the mean responses for the treatments arethe same.� B. We can reject the null hypothesis that the mean re-sponses for the treatments are the same and accept thealternative hypothesis that at least two treatment meansdiffer.

(c) Sum of Squares for Treatment (SST) is 90% of SS(Total)F �Rejection region F "

The final conclustion is1

� A. We can reject the null hypothesis that the mean re-sponses for the treatments are the same and accept thealternative hypothesis that at least two treatment meansdiffer.� B. There is not sufficient evidence to reject the null hy-pothesis that the mean responses for the treatments arethe same.

6.(1 pt) Use the Minitab display to test the claims. Use theα � 0 � 05 significance level. The sample data are SAT scores onthe verbal and math portions of SAT-I.

Analysis of Variance for SATSource DF SS MS F PGender 1 52831 52831 5 � 05 0 � 031Ver/Math 1 6023 6023 0 � 58 0 � 453Interaction 1 31630 31630 3 � 02 0 � 091Error 37 377494 10464Total 40 467978

Test the claim that SAT scores are not affected by an interac-tion between gender and test (verbal/math).

The F � test statistic isThe P-value isDoes there appear to be a significant effect from the interac-

tion between gender and test?� A. Yes� B. No

Test the claim that gender has an effect on SAT scores.The F � test statistic isThe P-value isIs there sufficient evidence to support the claim that gender

has an effect on SAT scores?� A. Yes� B. No

Test the claim that the type of test (math/verbal) has an effecton SAT scores.

The F � test statistic isThe P-value isIs there sufficient evidence to support the claim that the type

of test has an effect on SAT scores?� A. No� B. Yes7.(1 pt) A study was conducted to see how people reacted to

certain facial expressions. A sample group of n � 36 was ran-domly divided into six groups. Each group was assigned to toview one picture of a person making a facial expression. Eachgroup saw a different picture, and the different expressions were(1) Surprised (2) Nervous (3) Scared (4) Sad (5) Excited (6) An-gry. After viewing the pictures, the subjects were asked to rank

the degree of dominance they inferred from the facial expressionthey saw. (The scale ranged from -10 to 10) The data collectedis summarized in the table below.

Surprised Nervous Scared Sad Excited Angry1 � 1 � 3 � 0 � 6 � 0 � 9 0 � 5 1 � 4� 1 � 2 � 1 � 9 � 1 � 7 � 0 � 9 0 � 2 � 1 � 9

1 � 5 1 � 2 � 0 � 4 � 0 � 4 � 1 � 5 � 1 � 7� 0 � 7 1 � 6 � 0 � 5 � 0 � 4 � 1 � 5 1 � 6� 1 � 8 � 0 � 0999999999999999 1 � 3 � 2 1 � 9 � 1 � 42 1 � 4 � 2 � 1 � 9 1 � 5 0 � 7

Complete the following ANOVA table

Source df SS MS FExpressions

ErrorTotal

8.(1 pt) Use the Minitab display to test the claims using thesignificance level of α � 0 � 05. The sample data are the numbersof support beams manufactured by 4 different operators using5 different machines. Assume that there is no interaction effectfrom operator and machine.

Analysis of Variance for BeamsSource DF SS MS F POperator 3 59 � 98 19 � 56 2 � 35 0 � 124Machine 4 92 � 89 46 � 22 5 � 55 0 � 013Error 12 47 � 93 8 � 33Total 19 200 � 8

Test the claim that the four operators have the same meanproduction output.

The F � test statistic isThe P-value isIs there sufficient evidence to warrant the rejection of the

claim that the four machine operators have the same mean pro-duction output?� A. No� B. Yes

Test the claim that the choice of machine has no effect on theproduction output.

The F � test statistic isThe P-value isIs there sufficient evidence to warrant the rejection of the

claim that the choice of machine has no effect on the produc-tion output?� A. No� B. Yes


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