Question

which score indicates the highest relative position? round your answer to two decimal places, if necessary.
(a) a score of 3.2 on a test with \\( \overline { x } = 4.1 \\) and \\( s = 1.4 \\).
(b) a score of 630 on a test, with \\( \overline { x } = 760 \\) and \\( s = 190 \\).
(c) a score of 41 on a test with \\( \overline { x } = 51 \\) and \\( s = 5 \\).
the score with the highest relative position is select \\( \checkmark \\), since the select \\( \checkmark \\) = \\( \square \\) is highest.

Explanation:

Step1: Calculate z - score for (a)

The formula for the z - score is \( z=\frac{x-\bar{x}}{s} \).
For (a): \( x = 3.2\), \(\bar{x}=4.1\), \(s = 1.4\)
\( z_{a}=\frac{3.2 - 4.1}{1.4}=\frac{- 0.9}{1.4}\approx - 0.64\)

Step2: Calculate z - score for (b)

For (b): \(x = 630\), \(\bar{x}=760\), \(s = 190\)
\( z_{b}=\frac{630 - 760}{190}=\frac{-130}{190}\approx - 0.68\)

Step3: Calculate z - score for (c)

For (c): \(x = 41\), \(\bar{x}=51\), \(s = 5\)
\( z_{c}=\frac{41 - 51}{5}=\frac{-10}{5}=-2.00\)

Since \(|z_{c}|>|z_{b}|>|z_{a}|\), and the z - score with the largest magnitude (in the negative case, we consider the distance from 0) for relative position (assuming we are comparing the positions in terms of how many standard deviations from the mean, and a more extreme negative z - score still indicates a distinct relative position compared to the others in terms of the formula - based ranking), (c) has the highest relative position (in the sense of being the most extreme among these three values when considering the z - score formula application for relative standing).

Answer:

(c), \( z = -2.00 \)