Question

Shoes Refer to Exercise 1. Jackson, who reported owning 22 pairs of shoes, has a standardized score of z=1.10.

a. Interpret this z-score.

b. The standard deviation of the distribution of the number of pairs of shoes owned in this sample of 20 boys is 9.42. Use this information along with Jackson’s z-score to find the mean of the distribution.

Short answer

Part (a) Jackson owns $22$pairs of shoes, which is $1.10$standard deviations more than the mean number of pairs owned by the other students.

Part (b) Mean of the distribution,$\mu =11.638$

Step-by-step solution

Part (a) Step 1: Given information

Value,$x=22$

Standard deviation, $\sigma =9.42$

$z-$ score,$z=1.10$

Part (a) Step 2: Concept

The formula used:$z=\frac{(value-mean)}{(s\tan darddeviation)}$

Part (a) Step 3: Explanation

The amount of standard deviations a value deviates from the mean is described by the $z-$score.

A value below the mean is indicated by a negative $z-$score.

Whereas,

The value above the mean is indicated by a positive $z-$score.

Thus,

Jackson owns $22$ pairs of shoes, which is $1.10$standard deviations more than the mean number of pairs owned by the other students, according to the sample.

Part (b) Step 1: Calculation

Calculate the z − score:

$z=\frac{x-\mu }{\sigma }$

Substitute values,

$1.10=\frac{22-\mu }{9.42}$

Multiply both sides by 9.42:

$10.362=22-\mu$

That becomes

$\mu =22-10.362=11.638$

Because the data values and the mean units are the same.

Thus,

The mean is $11.638$ pairs of shoes.