mrs. lerner charlotte catholic high school with some updates by dr. davidson mallard creek high...
TRANSCRIPT
![Page 1: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/1.jpg)
Mrs. LernerCharlotte Catholic High School
With some updates byDr. Davidson
Mallard Creek High School
![Page 2: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/2.jpg)
1. Are uncertain in the short run2. Exhibit a consistent pattern in the
long run Note the dual aspect
Note: This is the Condition for the Probability we study
![Page 3: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/3.jpg)
An event is an outcome or a set of outcomes of random phenomenon (RP).
Outcome(s) of interest are “Success(es)” [even if they are bad news!!!!]
![Page 4: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/4.jpg)
(note dual aspect) Probability of an event =
1.proportion of times a success occurs in the long run (measure by experiment or simulation)
2.[number of ways for a success to occur]/[all possible events (specified by a model of the RP)
![Page 5: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/5.jpg)
If there are n ways to do a first event & m ways to do a second event Then the number of all possible outcomes=nm
This is called the multiplication rule
![Page 6: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/6.jpg)
Consider the sum of two dice that are rolled
Let’s say we are interested in the Probability the sum is a 4, so SUCCESS = ?
Find # of ways to get a sum = 4 [a success]is 3: 1+3, 2+2, 3+1 [3 ways]
![Page 7: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/7.jpg)
The probability of a specific outcome is [# of ways to get a success]/ [all
possible outcomes e.g. Probability of getting a sum =4 is: 3/36 = 1/12 ~ .0833 = 8.33 %
1. This is the chance the next outcome is a success
2. This is the proportion of times that successes occur in a large repetitive # of identical trials
![Page 8: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/8.jpg)
The probability P(A) of any event A satisfies 0≤P(A)≤1
Thus the probability of any event is between 0 & 1
![Page 9: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/9.jpg)
The Sample Space is the set of all possible outcomes
If S is the sample space in a probability model, then P(S)=1
The probabilities of all possible outcomes must add up to 1
![Page 10: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/10.jpg)
The complement of any event A is
1. the event that A does not occur
2. written as Ac
The complement rule states that
P(Ac)=1-P(A)The probability that an event
does not occur is 1 minus the probability that it does occur
![Page 11: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/11.jpg)
Two events A and B are disjoint 1. if they have no outcomes in
common 2. cannot occur at the same time.
If A and B are disjoint, then
P(A or B) = P(A) + P(B).If two events have no outcomes in common, then the probability of
either one occurring is the sum of their individual probabilities
![Page 12: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/12.jpg)
For any two events A and B:
P(A or B) = P(A) + P(B) – P(A and B)
The probability of either A or B occurring is the sum of their individual probabilities minus the probability that they occur at the same time
![Page 13: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/13.jpg)
Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent, then P(A and B) = P(A)P(B).
If two events are independent, then the probability that they both occur is the product of their individual probabilities
Note – Disjoint events are always NOT INDEPENDENT, for if one occurs, then we know the other can not occur!
![Page 14: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/14.jpg)
And = Joint = Intersection Or = Both (1, or other, or 2) = Union
and B
or
P A P A B
P A B P A B
![Page 15: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/15.jpg)
![Page 16: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/16.jpg)
For any two events A and B,
P(A and B) = P(B|A) P(A)P(A and B) = P(A|B) P(B)
The probability of A and B occurring jointly is the probability that B occurs given that A has already occurred multiplied by the probability that A occurs
![Page 17: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/17.jpg)
Example - 2:Suppose that for a certain Caribbean island
in any 3-year period:1. the probability of a major hurricane
is .252. the probability of water damage is .44
3. and the probability of both a hurricane and water damage is .22.
What is the probability of water damage given that there is a hurricane?
![Page 18: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/18.jpg)
Suppose that for a certain Caribbean island in any 3-year period the probability of a major hurricane is .25, the probability of water damage is .44 and the probability of both a hurricane and water damage is .22. What is the probability of water damage given that there is a hurricane?
water damage and hurricanewater damage|hurricane
hurricane
.22
.25.88
PP
P
![Page 19: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/19.jpg)
If three people, Joe, Betsy, and Sue, play a game in which
Joe has a 25% chance of winning Betsy has a 40% chance of
winning What is the probability that Sue
will win?
![Page 20: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/20.jpg)
2. If three people, Joe, Betsy, and Sue, play a game in which Joe has a 25%
chance of winning and Betsy has a 40% chance of winning, what is the probability that Sue will win?
Sue 1 .25 .4
1 .65
.35
P
![Page 21: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/21.jpg)
4. A summer resort rents rowboats to customers but does not allow more than four people to a boat. Each boat is designed to hold no more than 800 pounds.
Suppose the distribution of the weight of adult males who rent boats, including their clothes and gear, is normal with a mean of 190 pounds and standard deviation of 10 pounds.
If the weights of individual passengers are independent, what is the probability that a group of four adult male
passengers will exceed the acceptable weight limit of 800 pounds?
![Page 22: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/22.jpg)
190
10X
X
4
4 190
760
T X
![Page 23: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/23.jpg)
190
10X
X
4
4 190
760
T X
2 2 2 2 2
2 2 2 210 10 10 10
400
20
T X X X X
T
![Page 24: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/24.jpg)
190
10X
X
4
4 190
760
T X
2 2 2 2 2
2 2 2 210 10 10 10
400
20
T X X X X
T
exceed 800 lbs 0.023P
800,1 99,760,20normalcdf E
![Page 25: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/25.jpg)
The following table shows the frequencies of political affiliations in the age ranges listed from a random sample of adult citizens in a particular city.
Dem. Repub. Indep.18-30 25 18 1231-40 32 21 1041-50 17 25 17Over 50 14 32 15
_____________________________________________________________
5. What proportion of the Republicans are over 50?
a. 61/238b. 32/96c. 96/238d. 32/61e. Cannot be determined.
![Page 26: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/26.jpg)
The following table shows the frequencies of political affiliations in the age ranges listed from a random sample of adult citizens in a particular city.
Dem. Repub. Indep.18-30 25 18 1231-40 32 21 1041-50 17 25 17Over 50 14 32 15
_____________________________________________________________
5. What proportion of the Republicans are over 50?
a. b. 32/96c. d. e.
![Page 27: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/27.jpg)
The following table shows the frequencies of political affiliations in the age ranges listed from a random sample of adult citizens in a particular city.
Dem. Repub. Indep.18-30 25 18 1231-40 32 21 1041-50 17 25 17Over 50 14 32 15
_____________________________________________________________
6. If one adult citizen is chosen at random, what is the probability that this person is
a Democrat between the ages of 41 and 50?
a. 17/238b. 17/88c. 61/238d. 17/61e. 88/238
![Page 28: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/28.jpg)
The following table shows the frequencies of political affiliations in the age ranges listed from a random sample of adult citizens in a particular city.
Dem. Repub. Indep.18-30 25 18 1231-40 32 21 1041-50 17 25 17Over 50 14 32 15
_____________________________________________________________
6. If one adult citizen is chosen at random, what is the probability that this person is
a Democrat between the ages of 41 and 50?
a. 17/238b. c. d. e.
![Page 29: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/29.jpg)
The following table shows the frequencies of political affiliations in the age ranges listed from a random sample of adult citizens in a particular city.
Dem. Repub. Indep.18-30 25 18 1231-40 32 21 1041-50 17 25 17Over 50 14 32 15
_____________________________________________________________
7. Given that a person chosen at random is between 31 and 40, what is the
probability that this person is an Independent?
a. 10/238b. 10/63c. 10/54d. 54/238e. 63/238
![Page 30: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/30.jpg)
The following table shows the frequencies of political affiliations in the age ranges listed from a random sample of adult citizens in a particular city.
Dem. Repub. Indep.18-30 25 18 1231-40 32 21 1041-50 17 25 17Over 50 14 32 15
_____________________________________________________________
7. Given that a person chosen at random is between 31 and 40, what is the
probability that this person is an Independent?
a. b. 10/63c. d. e.
![Page 31: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/31.jpg)
The following table shows the frequencies of political affiliations in the age ranges listed from a random sample of adult citizens in a particular city.
Dem. Repub. Indep.18-30 25 18 1231-40 32 21 1041-50 17 25 17Over 50 14 32 15
_____________________________________________________________
8. What proportion of the citizens sampled are over 50 or Independent?
a. 54/238b. 61/238c. 100/238d. 115/238e. Cannot be determined
![Page 32: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/32.jpg)
The following table shows the frequencies of political affiliations in the age ranges listed from a random sample of adult citizens in a particular city.
Dem. Repub. Indep.18-30 25 18 1231-40 32 21 1041-50 17 25 17Over 50 14 32 15
_____________________________________________________________
8. What proportion of the citizens sampled are over 50 or Independent?
a. b. c. 100/238d. e.
![Page 33: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/33.jpg)
Example - Assume an multiple-choice examination
consists of questions, each having five possible answers.
Linda estimates that she has probability 0.75 of knowing the answer to any question that may be asked.
If she does not know the answer, she will guess, with conditional probability 1/5 of being correct. What is the probability that Linda gives the correct answer to a question? (Draw a tree diagram to guide the calculations.)
![Page 34: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/34.jpg)
An examination consists of multiple-choice questions, each having five possible answers. Linda estimates that she has probability 0.75 of knowing the answer to any question that may be asked. If she does not know the answer, she will guess, with conditional probability 1/5 of being correct. What is the probability that Linda gives the correct answer to a question? (Draw a tree diagram to guide the calculations.)
P(correct) = .75 +.25*.2 = .8
![Page 35: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/35.jpg)
A random variable assumes any of several different values as a result of some random phenomenon
![Page 36: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/36.jpg)
A Discrete RV – has a countable number of possible values
◦ The probabilities must satisfy two requirements Every probability is between 0 and 1 The sum of all probabilities is 1
◦ We can use a probability histogram to look at the probability distribution.
Value of X
x1 x2 x3 … xk
Probability
p1 p2 p3 … pk
![Page 37: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/37.jpg)
Mean of a Discrete R. V. – (also called expected value) –
ii
kkX
px
pxpxpxpx
332211
![Page 38: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/38.jpg)
Variance of a Discrete R. V. –
iXi
kXkXXXX
px
pxpxpxpx2
23
232
221
21
2
![Page 39: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/39.jpg)
Continuous R. V. – takes all values in an interval of numbers
◦We look at its distribution using a density curve
◦The probability of any event is the area under the density curve in that interval.
![Page 40: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/40.jpg)
1. If X is an R. V. and a & b are fixed numbers, then themean μa+bX = a +bμX
2. If X and Y are R. V.‘s, then μX±Y = μX ± μY
![Page 41: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/41.jpg)
If X is an R. V. and a and b are fixed numbers, then σ2
a+bX = b2σX2
◦ Note that multiplying by a constant changes the variance but adding a constant does not.
If X and Y are independent R. V.’s, then σ2
X±Y =σX2+σY
2
“Pythagorean Theorem of Statistics”
For STANDARD DEVIATION:square ‘em, add ‘em, take the square root
![Page 42: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/42.jpg)
11. Suppose X and Y are random variables with μX = 10, σX = 3, μY = 15, and
σY = 4. Given that X and Y are independent, what are the mean and standard deviation of the random variable X+Y?
![Page 43: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/43.jpg)
11. Suppose X and Y are random variables with μX = 10, σX = 3, μY = 15, and
σY = 4. Given that X and Y are independent, what are the mean and standard deviation of the random variable X+Y?
μX+Y = μX + μY σX+Y = √σ2X + σ2
Y
=10 + 15 = √9+16= 25 = √25
= 5
![Page 44: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/44.jpg)
12. You roll a die. If it comes up a 6, you win $100. If not, you get to roll again. If you get a 6 the second time, you win $50. If not, you lose.
a. Create a probability model for the amount you
will win at this game.
![Page 45: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/45.jpg)
12. You roll a die. If it comes up a 6, you win $100. If not, you get to roll again. If you get a 6 the second time, you win $50. If not, you lose.
a. Create a probability model for the amount you
will win at this game.Winnings $100 $50 $0
Probability 1/6 (5/6)(1/6)5/36
(5/6)(5/6)25/36
![Page 46: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/46.jpg)
12. You roll a die. If it comes up a 6, you win $100. If not, you get to roll again. If you get a 6 the second time, you win $50. If not, you lose.
b. Find the expected amount you’ll win.
![Page 47: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/47.jpg)
12. You roll a die. If it comes up a 6, you win $100. If not, you get to roll again. If you get a 6 the second time, you win $50. If not, you lose.
b. Find the expected amount you’ll win.
1 5 25
100 50 06 36 26
$23.61
X E X
![Page 48: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/48.jpg)
Law of Large Numbers – The long run relative frequency of repeated independent trials gets closer and closer to the true relative frequency as the number of trials increases.
![Page 49: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/49.jpg)
Binomial Distribution – the distribution of the count X successes in the binomial setting.
B(n,p) where n is the number of
observations and p is the probability of success
![Page 50: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/50.jpg)
Use binompdf(n,p,X) to find the probability of a single value of X, such as P(X = 3).
Use binomcdf(n,p,X)to find the probability of at most X successes
for example P(X ≤ 3).
![Page 51: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/51.jpg)
pnp
np
1
![Page 52: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/52.jpg)
13. Pepsi is running a sales promotion in which 12% of all bottles have a “FREE”
logo under the cap. What is the probability that you find two free cans in a 6-
pack?
![Page 53: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/53.jpg)
13. Pepsi is running a sales promotion in which 12% of all bottles have a “FREE”
logo under the cap. What is the probability that you find two free cans in a 6-
pack? 2 46
2 .12 .882
P X
![Page 54: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/54.jpg)
13. Pepsi is running a sales promotion in which 12% of all bottles have a “FREE”
logo under the cap. What is the probability that you find two free cans in a 6-
pack?
2 462 .12 .88
2P X
6,.12,2binompdf
2 .13P X
![Page 55: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/55.jpg)
14. The National Association of Retailers reports that 62% of all purchases are now made by credit card; you think
this is true at your store as well. On a typical day you make 20 sales.
a. Let X represent the number of customers who use a credit card on a typical day. What is the probability model for X?
![Page 56: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/56.jpg)
14. The National Association of Retailers reports that 62% of all purchases are now made by credit card; you think
this is true at your store as well. On a typical day you make 20 sales.
a. Let X represent the number of customers who use a credit card on a typical day. What is the probability model for X?
The model is = B(20, .62) [i.e. B(n,p)] Please Explain Why
![Page 57: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/57.jpg)
14. The National Association of Retailers reports that 62% of all purchases are now made by credit card; you think this is true at your store as well. On a typical day you make 20 sales.
b. Find the mean and standard deviation.
![Page 58: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/58.jpg)
14. The National Association of Retailers reports that 62% of all purchases are now made by credit card; you think this is true at your store as well. On a typical day you make 20 sales.
b. Find the mean and standard deviation.
20 .62
12.4
X np
1
12 .62 .38
2.17
X np p
![Page 59: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/59.jpg)
14. The National Association of Retailers reports that 62% of all purchases are now made by credit card; you think this is true at your store as well. On a typical day you make 20 sales.
c. What is the probability that on a typical day at least half of your customers
use a credit card?
![Page 60: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/60.jpg)
14. The National Association of Retailers reports that 62% of all purchases are now made by credit card; you think this is true at your store as well. On a typical day you make 20 sales.
c. What is the probability that on a typical day at least half of your customers
use a credit card? 10 1 9
.9077
P X P X
1 (20,.62,9)binomcdf
![Page 61: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/61.jpg)
19. The volumes of soda in quart soda bottles can be described by a Normal model with a mean of 32.3 oz and a standard deviation of 1.2 oz. What is the probability that a randomly selected bottle has a volume less than 32 oz?
![Page 62: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/62.jpg)
There are typos in the next slide: The z score calculation should read:
P(x< 32) = P(z < [32-32.3]/1.2) =P(z< -.25) = .4013
AndNormalcdf(-E99, 32, 32.3, 1.2) = .4013
![Page 63: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/63.jpg)
19. The volumes of soda in quart soda bottles can be described by a Normal model
N(32.3, 1.2) a mean of 32.3 oz and a standard deviation of 1.2 oz.
What is the probability that the volume of a randomly selected bottle has a less than 32 oz?
32.3 3232
1.2
.1429
.4013
P x P z
P z
( 1 99,32,32.1,1.2)normalcdf E
![Page 64: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/64.jpg)
20. A bank's loan officer rates applicants for credit. The ratings can be described by a Normal model with a mean of 200 and a standard deviation of 50. If an applicant is randomly selected, what is the probability that the rating is between 200 and 275?
![Page 65: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/65.jpg)
20. A bank's loan officer rates applicants for credit. The ratings can be described by a Normal model with a mean of 200 and a standard deviation of 50. If an applicant is randomly selected, what is the probability that the rating is between 200 and 275?
200 275 .4332P x
![Page 66: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/66.jpg)
Sampling distribution – the distribution of values taken by a statistic in all possible samples of the same size from the same population
![Page 67: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/67.jpg)
Provided that the sampled values are independent and the sample size is large enough, the sampling distribution of is modeled by a Normal model with meanand standard deviation .
p̂
p̂ p
1ˆ
p pSD p
n
![Page 68: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/68.jpg)
Assume that 12% of students at a university wear contact lenses. We randomly pick 200 students. ◦ What is the mean of the proportion of students in
this group who may wear contact lenses? ◦ What is the standard deviation of the proportion
of students in this group who may wear contact lenses?
![Page 69: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/69.jpg)
Assume that 12% of students at a university wear contact lenses. We randomly pick 200 students. ◦ What is the mean of the proportion of students in
this group who may wear contact lenses? ◦ What is the standard deviation of the proportion
of students in this group who may wear contact lenses?
.12
.12 .88
200.023
![Page 70: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/70.jpg)
Suppose that x-bar is the mean of an SRS of size n drawn from a large population with mean μ and standard deviation σ.
Then the◦ mean of the sampling distribution of xbar is μ
(hence xbar is an unbiased indicator of μ)◦ standard deviation of the sampling distribution of
xbar is
σ /√n.
![Page 71: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/71.jpg)
The scores of individual students on the ACT have a normal distribution with mean 18.6 and standard deviation 5.9. At Northside High, 76 seniors take the test. If the scores at this school have the same distribution as national scores, what are the mean and standard deviation of the distribution of sample means for these 76 students?
![Page 72: Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School](https://reader035.vdocuments.net/reader035/viewer/2022062511/551bd82b550346b4588b589b/html5/thumbnails/72.jpg)
The scores of individual students on the ACT have a normal distribution with mean 18.6 and standard deviation 5.9. At Northside High, 76 seniors take the test. If the scores at this school have the same distribution as national scores, what are the mean and standard deviation of the average (sample mean) for the 76 students?
18.6
5.9
76.6768