Question

how many pairs of shoes does a typical teenage boy own? to find out, two ap statistics students surveyed a random sample of 20 male students from their large high school. then they recorded the number of pairs of shoes that each boy owned. given is a dot - plot of the data. jackson, who reported owning 22 pairs of shoes, has a standardized score of z = 1.10. the standard deviation of the distribution of number of pairs of shoes owned in this sample of 20 boys is 9.42. what is the mean of the distribution? (\bar{x}=11.64) pairs (\bar{x}=81.62) pairs (\bar{x}=20) pairs of shoes (\bar{x}=8.16) pairs

Explanation:

Step1: Recall the z - score formula

The z - score formula is $z=\frac{x-\bar{x}}{s}$, where $z$ is the standardized score, $x$ is the individual value, $\bar{x}$ is the mean, and $s$ is the standard deviation.
We know that $x = 22$ (Jackson's number of shoes), $z=1.10$, and $s = 9.42$.

Step2: Rearrange the z - score formula to solve for the mean

Starting with $z=\frac{x-\bar{x}}{s}$, we can multiply both sides by $s$: $z\times s=x - \bar{x}$.
Then, we can rewrite it to solve for $\bar{x}$: $\bar{x}=x - z\times s$.

Step3: Substitute the known values into the formula

Substitute $x = 22$, $z = 1.10$, and $s=9.42$ into the formula $\bar{x}=x - z\times s$.
$\bar{x}=22-1.10\times9.42$.
First, calculate $1.10\times9.42 = 10.362$.
Then, $\bar{x}=22 - 10.362=11.638\approx11.64$.

Answer:

$\bar{x}=11.64$ pairs, so the correct option is $\bar{x}=11.64$ pairs.