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, since the select \\( < \\) is highest.

Explanation:

Step1: Calculate z - scores

Use the formula $z=\frac{x-\overline{X}}{s}$ for each case.
For (a): $z=\frac{3.2 - 4.1}{1.4}$
For (b): $z=\frac{630 - 760}{190}$
For (c): $z=\frac{41 - 51}{5}$

Step2: Compare z - scores

Compare the calculated z - scores. Higher z - score (less negative) indicates higher relative position.

Answer:

A score of 3.2 on a test with $\overline{X}=4.1$ and $s = 1.4$: $z=\frac{3.2 - 4.1}{1.4}=\frac{- 0.9}{1.4}\approx - 0.64$

A score of 630 on a test with $\overline{X}=760$ and $s = 190$: $z=\frac{630 - 760}{190}=\frac{-130}{190}\approx - 0.68$

A score of 41 on a test with $\overline{X}=51$ and $s = 5$: $z=\frac{41 - 51}{5}=\frac{-10}{5}=-2$

Since $-0.64>-0.68 > - 2$, the score in part (a) has the highest relative position.