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A large university administers entrance exams to all entering students. The test is administered again as an exit exam for all graduates. The test results for one group of students are given in the table. Freshman Year Senior Year X ¯ = 53 X ¯ = 92 s = 7 s = 4 a. The scores for five students on both tests are listed below. For each student, calculate Z scores and determine whether the student performed better as a freshman or as a senior. Student Score, Freshman Year Score, Senior Year A 57 97 B 51 94 C 45 82 D 73 101 E 62 98 b. Determine the probabilities that randomly selected students will have scores in each of these ranges: Freshman Year Score Probability Less than 52 Less than 57 Between 40 and 50 More than 51 More than 62 Senior Year Score Probability Less than 88 Less than 98 Between 80 and 100 More than 97 More than 85

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Essentials Of Statistics

4th Edition
HEALEY + 1 other
Publisher: Cengage Learning,
ISBN: 9781305093836
BuyFind

Essentials Of Statistics

4th Edition
HEALEY + 1 other
Publisher: Cengage Learning,
ISBN: 9781305093836

Solutions

Chapter
Section
Chapter 5, Problem 5.15P
Textbook Problem
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A large university administers entrance exams to all entering students. The test is administered again as an exit exam for all graduates. The test results for one group of students are given in the table.

Freshman Year Senior Year
X ¯ = 53 X ¯ = 92
s = 7 s = 4

a. The scores for five students on both tests are listed below. For each student, calculate Z scores and determine whether the student performed better as a freshman or as a senior.

Student Score, Freshman Year Score, Senior Year
A 57 97
B 51 94
C 45 82
D 73 101
E 62 98

b. Determine the probabilities that randomly selected students will have scores in each of these ranges:

Freshman Year
Score Probability
Less than 52
Less than 57
Between 40 and 50
More than 51
More than 62
Senior Year
Score Probability
Less than 88
Less than 98
Between 80 and 100
More than 97
More than 85

Expert Solution
To determine

(a)

To find:

The Z scores and to determine whether the student performed better as a freshman or as a senior.

Explanation of Solution

Given:

A large university administers entrance exams to all entering students. The test is administered again as an exit exam for all graduates. The test results for one group of students are given in the table below,

Freshman Year Senior Year
X¯=53 X¯=92
s=7 s=4

Description:

The normal curve is symmetrical and its mean is equal to the median. The area below and above the mean is 50% or 0.05.

Formula used:

Let the data values be denoted by Xi.

The formula to calculate the Z score is given by,

Z=XiX¯s

Where, X¯ is the mean and s is the standard deviation of the distribution.

Calculation:

Given that for freshmen year the mean is 53 and standard deviation is 7.

For the score of 57,

Substitute 57 for Xi, 53 for X¯, and 7 for s in the above mentioned formula,

Z=57537=47=0.57

For the score of 51,

Substitute 51 for Xi, 53 for X¯, and 7 for s in the above mentioned formula,

Z=51537=27=0.29

For the score of 45,

Substitute 45 for Xi, 53 for X¯, and 7 for s in the above mentioned formula,

Z=45537=87=1.14

For the score of 73,

Substitute 73 for Xi, 53 for X¯, and 7 for s in the above mentioned formula,

Z=73537=207=2.86

For the score of 62,

Substitute 62 for Xi, 53 for X¯, and 7 for s in the above mentioned formula,

Z=62537=97=1.29

Given that for senior year the mean is 92 and standard deviation is 4.

For the score of 97,

Substitute 97 for Xi, 92 for X¯, and 4 for s in the above mentioned formula,

Expert Solution
To determine

(b)

To find:

The probabilities of the given ranges.

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