Question

a random sample of 15 of the 75 competitors at a middle school gymnastics competition are asked their height. the data set lists the heights in inches: 52, 55, 58, 59, 59, 59, 59, 59, 59, 60, 61, 63, 66, 66, 67. what is the mean height of the sample? if necessary, round your answer to the nearest tenth. could you say this is a good estimate of the mean height of all competitors? why or why not? the mean height of the 15 gymnasts in the sample is \boxed{} inches. this \boxed{} a good estimate of the mean height of all the competitors because the sample is \boxed{} (select) downarrow.

Explanation:

Step1: Sum all the heights

We have the heights: 52, 55, 58, 59, 59, 59, 59, 59, 60, 61, 63, 66, 66, 66, 67.
Let's calculate the sum:
\[

$$\begin{align*} &52 + 55 + 58 + 59\times5 + 60 + 61 + 63 + 66\times3 + 67\\ =&52 + 55 + 58 + 295 + 60 + 61 + 63 + 198 + 67\\ =&(52 + 55) + 58 + 295 + 60 + 61 + 63 + 198 + 67\\ =&107 + 58 + 295 + 60 + 61 + 63 + 198 + 67\\ =&165 + 295 + 60 + 61 + 63 + 198 + 67\\ =&460 + 60 + 61 + 63 + 198 + 67\\ =&520 + 61 + 63 + 198 + 67\\ =&581 + 63 + 198 + 67\\ =&644 + 198 + 67\\ =&842 + 67\\ =&909 \end{align*}$$

\]

Step2: Calculate the mean

The mean is the sum divided by the number of data points (n = 15).
\[
\text{Mean} = \frac{909}{15} = 60.6
\]
Now, for the second part: A random sample of 15 out of 75 competitors. Since the sample is random, it should be representative of the population. So the mean height of the sample (60.6 inches) is a good estimate of the mean height of all competitors because the sample is randomly selected, which should make it representative of the population.

Answer:

The mean height of the 15 gymnasts in the sample is $\boldsymbol{60.6}$ inches. This $\boldsymbol{\text{is}}$ a good estimate of the mean height of all the competitors because the sample is randomly selected, so it should be representative of the population.