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Chapter 4: Problem 43

A student took two national aptitude tests. The national average and standard deviation were 475 and 100 , respectively, for the first test and 30 and 8 , respectively, for the second test. The student scored 625 on the first test and 45 on the second test. Use \(z\) scores to determine on which exam the student performed better relative to the other test takers.

Short Answer

Expert verified
The student performed better on the second exam compared to the first, in relation to other test takers.

Step by step solution

01

Calculating the Z score for the First Test

We start by calculating the \(z\) score for the first test using the formula: \(z = (X - 渭) / 蟽\), where \(X\) is the student's score, \(渭\) is the mean, and \(蟽\) is the standard deviation. For the first test, \(X = 625, 渭 = 475\), and \(蟽 = 100\). Substituting these gives \(z = (625 - 475) / 100 = 1.5\).
02

Calculating the Z score for the Second Test

We do the same for the second test. In this case, \(X = 45, 渭 = 30\), and \(蟽 = 8\). Substituting these we get \(z = (45 - 30) / 8 = 1.875\).
03

Comparing the Z scores

Now that we have the two \(z\) scores, we can conclude. A higher \(z\) score means the student did better compared to other test takers. In this case, the \(z\) score for the second test, 1.875, is higher than the \(z\) score for the first test, 1.5. Thus, the student did better on the second test compared to other test takers.

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