Question

1. 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 number of pairs of shoes owned in this sample of 20 boys is 9.42. use this information along with jackson’s standardized score to find the mean of the distribution.

Explanation:

Step1: Recall z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the data - point, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Step2: Interpret z - score

A z - score of $z = 1.10$ means that Jackson's number of shoes (22 pairs) is 1.10 standard deviations above the mean number of shoes owned by the 20 boys in the sample.

Step3: Use z - score formula to find the mean

We know that $z = 1.10$, $x = 22$, and $\sigma=9.42$. Substitute these values into the z - score formula $z=\frac{x - \mu}{\sigma}$ and solve for $\mu$.
$1.10=\frac{22-\mu}{9.42}$
Multiply both sides by 9.42: $1.10\times9.42=22-\mu$
$10.362 = 22-\mu$
Add $\mu$ to both sides: $\mu+10.362 = 22$
Subtract 10.362 from both sides: $\mu=22 - 10.362=11.638$

Answer:

(a) Jackson's number of shoes is 1.10 standard deviations above the mean number of shoes owned by the 20 boys in the sample.
(b) The mean of the distribution is 11.638.