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York University

MATH 2560 (FALL 2004): Elementary Statistics I

Mid Term Test 1 Tuesday October 12, 2004

Last Name: Given Names:

Student Number:

Signature :

DO NOT WRITE IN THIS AREA

Read the following instructions carefully

1. This exam is closed book and closed notes.

2. You are allowed a calculator. You may NOT borrow a calculator from another student.

3. All work must be done in the free space provided below for each problem or on the

back of the preceding page.

4. No other sheets of paper will be accepted or marked.

5. Show all work. Answers without sufficient supporting work will receive *no credit*,

even if the answers are correct.

6. Your final grade is

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Question 1 The average GMAT scores of students entering the 25 top-ranked US schools of busi-

ness were published. The data are as follows

School Av.GMAT score School Av.GMAT score

StanfordHarvard 644 Carnegie-Mello 620Penn 644 Yale 657

Northwestern 642 North Carolina 620

M.I.T. 650 New York Univ. 609

Chicago 635 Indiana 610

Michigan 630 Texas 631

Columbia 635 USC 606

Duke 630 Rochester 608

Dartmouth 643 Purdue 601

Virginia 617 Pittsburgh 597

Cornell 635 Vanderbilt 602U.C.Berkeley 635

a. 6 points Draw a histogram of the data using [595, 605) as the first class.

The number of average GMAT scores that fall in each class is as follows:

[595, 605) 3

[605, 615) 4

[615, 625) 3

[625, 635) 3

[635, 645) 8

[645, 655) 1

[655, 665) 1

b. 6 points Draw a stem and leaf plot of the data with leaf unit equal to 1.

59 7

60 12689

61 07

62 00

63 0015555

64 2344

65 07

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c. 6 points Ignore the units in the given data and draw a stem and leaf plot of the

rounded data with leaf unit equal to 10 and one line per 100. Is this stem and plot

diagram informative?

5 9

6 0000011223333333444455

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Question 2 Consider the data given in Question 1.

a. 6 points Find the median, first and third quartiles of this data set.

Q1 = 609 M= 630 Q3 = 642

b. 6 points Do you detect any outliers? Justify your answer.

IQR = 642 609 = 33 1.5IQR = 49.5

Q1 1.5IQR = 609 49.5 = 559.5

Q3 + 1.5IQR = 642 + 49.5 = 691.5

There are no points outside this range. No outliers according to the IQR rule.

c. 6 points Draw the box plot diagram of the data clearly indicating the different

points of interest.

MIN = 597 Q1 = 609 Q2 = 630 Q3 = 642 MAX= 657

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Question 3. The stem and leaf plot of 10 values Y1, . . . , Y 10 is as follows

Stem-and-leaf of C2 N = 10

Leaf Unit = 0.10

1 -1 5

1 -1

4 -0 776

5 -0 0

5 0 123

2 0

2 1 4

1 1 7

a. 6 points Compute the sample mean Y of the Yis.

Y = 1.50.70.70.6+0+0.1+0.2+0.3+1.4+1.710

= 0.02

b. 6 points Compute the sample mean of the Zi, i = 1, . . . , 10 ifZi = 2Yi + 10.

Z= 2(0.02) + 10 = 10 0.04 = 9.96

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Question 4 (6 points) A club has 30 student members and 10 faculty members. The students are

Abel Fisher Huber Moran Reinman

Carson Golomb Jimenez Moskowitz Santos

Chen Graiswold Jones neyman ShawDavid Hein Kiefer OBrien Thompson

Deming Heernandez Klotz Pearl Utts

Elashoff Holland Liu Potter Vlasic

and the faculty members are

Andrews Fernandez Kim Moore Rabinowitz

Besicovitch Gupta lightman Phillips Yang

The club decides to send four students and two faculty members to a convention.

They are chosen according to a simple random sample within each category, students

and faculty.

What sort of random sampling is this?

This is stratified sampling since there are two groups or strata and an SRS is done

within each of them.

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In the following problem, when reading the normal tables, if you cannot

find the exact z-score corresponding to a certain probability, take the av-

erage of the z-scores corresponding to the probabilities immediately below

and immediately above that probability .

Question 5 Suppose that your statistics professor returned your first mid term exam with only

a z-score (i.e. standardized variable) written on it. However she tells you that the

grades out of 100 follow a normal N(65, 10) distribution.

(a) 8 points What is your grade out of 100 if your z-score (i.e. standardized grade)

is 1.2 ?

Y65

10= 1.2 Y = 65 + 12 = 77

(b) 8 pointsWhich grade must you have in order to be in the top 10% of the class?

P(Z z) = 0.10 z = 1.285

Y65

10= 1.285 Y = 65 + 12.85 = 77.85

(c) 8 points What is the probability that the grade of a randomly chosen student in

the class is between 50 and 60?

P(50 Y 60) = P(506510

Z 606510

)

= P(1.5 Z 0.5) = P(Z 0.5) P(Z 1.5)

= 0.3085 0.0668 = 0.2417

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(d) 8 pointsWhat percentage of the class do you expect to be above the mean minus

two standard deviations but below the mean plus one standard deviation.

P( 2 Y + ) = 12

(0.95 + 0.68) = 0.815

81.5%

(e) 8 points What is the probability that your grade is either greater than 90 or lessthan 10?

P(Y 90) + P(Y 10) = P(Z 906510

) + P(Z 106510

)

= P(Z 2.5) + 0 = 1 0.9938 = 0.0062

Question 6 (6 points) A small class has a population of 11 students listed below.

Label Name Gender

1 Luke Male

2 John Male

3 Ryan Male

4 Travoris Male

5 Michael Male

6 Alicia Female

7 William Male

8 Charles Male

9 Bhavini Female

10 Geoferey Male

11 Nicole Female

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Use the following list of random digits

03802 29341 29264 80698 13370 13121 54969 43942 77320 35030 77519 41109

starting at the beginning of this list, to choose a simple random sample of four

students. Use the labels attached to the 11 names. The estimate of the population

proportion of female students in this class based on the selected sample is

(A) 0.00

(B) 0.18

(C) 0.25

(D) 0.50

(E) 0.75

The chosen labels are 03, 06, 01 and 11 corresponding to Ryan, Alicia, Luke and Nicole.

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