Question

Suppose that scores on a knowledge test are normally distributed with a mean of 60 and a standard deviation of 4.3. Scores on an aptitude test are normally distributed with a mean of 110 and a standard deviation of 7.1.

Boris scored a 57 on the knowledge test and 106 on the aptitude test. Callie scored 63 on the knowledge test and 114 on the aptitude test.


(a) Which test did Boris perform better on? Use z-scores to support your answer.

(b) Which test did Callie perform better on? Use z-scores to support your answer.

(c) Boris also took a logic test. His z-score on that test was . Does this change the answer to which test Boris performed better on? Explain your answer using z-scores.

Answer 1

Question:

(c) Boris also took a logic test. His z-score on that test was +0.93 . Does this change the answer to which test Boris performed better on? Explain your answer using z-scores.

Answer:

The answers to the questions are;

(a) Based on the z score, Boris perform better on his aptitude test.

(b) Based on the z score, Callie perform better on his knowledge test .

(c) For Boris since +0.93 =
`z_(logic)` >
`z_(knowledge)` >
`z_(aptitude)`

Yes as Boris now performed best on the logic test.

Explanation:

The z-score of a score is a measurement of the score withe respect to its distance from the mean as a factor of the standard deviation.

To solve the question, we note that we are required to find the z score as follows.

z score is given by
`z = (x -\mu)/(\sigma)`

Where:

z = Standard score

x = Score

σ = Standard deviation

μ = Mean

(a) To find out which test did Boris performed better usin z score, we have

Boris scored a

57 on the knowledge test and

106 on the aptitude test

Therefore the z sore for the knowledge test is

`z = (x -\mu)/(\sigma)`

Here

x = 57

μ = 60

σ = 4.3

Therefore

`z_(knowledge) = (57 -60)/(4.3)` = -3/4.3 = -30/43 = -0.6977

The z sore for Boris on the aptitude test is

Here

x = 106

μ = 110

σ = 7.1

`z_(aptitude) = (106 -110)/(7.1)` = -40/17 = -0.5634

Based on the z score, Boris perform better on the aptitude test as his z score is higher (on the number line), --0.5634, compared to the z score on the knowledge test , -0.6977

(b) For Callie we have

Callie scored a

63 on the knowledge test and

114 on the aptitude test

Therefore the z sore for the knowledge test is

`z = (x -\mu)/(\sigma)`

Here

x = 63

μ = 60

σ = 4.3

Therefore

`z_(knowledge) = (63 -60)/(4.3)` = 3/4.3 = 30/43 = 0.6977

The z sore for Callie on the aptitude test is

Here

x = 114

μ = 110

σ = 7.1

`z_(aptitude) = (114 -110)/(7.1)` = 40/17 = 0.5634

Based on the z score, Callie perform better on the knowledge test as his z score is higher (on the number line), 0.6977, compared to the z score on the aptitude test , 0.5634.

(c) If
`z_(logic)` = +0.93 then sinc for Boris
`z_(knowledge)` = -0.6977 and

`z_(aptitude)`= - 0.5634 then

`z_(logic)` >
`z_(knowledge)` >
`z_(aptitude)`

Therefore Boris now performed best on the logic test.