Every year, the students at Gossett High School take a...

Every year, the students at Gossett High School take a physical fitness test during their gym classes. One component of the test asks them to do as many push-ups as they can. Results for one class are shown below, separately for boys and girls. Assuming that students at Gossett are assigned to gym classes at random, create a $90 \%$ confidence interval for how many more push-ups boys can do than girls, on average, at that high school. $$\begin{array}{|l|l|r|l|l|r|r|r|r|r|r|r|r|} \hline \text { Boys } & 17 & 27 & 31 & 17 & 25 & 32 & 28 & 23 & 25 & 16 & 11 & 34 \\ \hline \text { Girls } & 24 & 7 & 14 & 16 & 2 & 15 & 19 & 25 & 10 & 27 & 31 & 8 \\ \hline\end{array}$$

Every year, the students at Gossett High School take a physical fitness test during their gym classes. One component of the test asks them to do as many push-ups as they can. Results for one class are shown below, separately for boys and girls. Assuming that students at Gossett are assigned to gym classes at random, create a $90 \%$ confidence interval for how many more push-ups boys can do than girls, on average, at that high school.
$$\begin{array}{|l|l|r|l|l|r|r|r|r|r|r|r|r|}
\hline \text { Boys } & 17 & 27 & 31 & 17 & 25 & 32 & 28 & 23 & 25 & 16 & 11 & 34 \\
\hline \text { Girls } & 24 & 7 & 14 & 16 & 2 & 15 & 19 & 25 & 10 & 27 & 31 & 8 \\
\hline\end{array}$$

Key Concepts

Confidence Interval

A confidence interval is a range of values used to estimate a population parameter, constructed so that there is a specified probability (in this case, 90%) that the interval contains the true value of the parameter. It provides not just a point estimate but also the uncertainty around that estimate, which is crucial in inferential statistics.

Difference of Means

This concept involves comparing the average values of two independent populations—in this context, the average number of push-ups by boys and girls. The goal is to infer the difference between the population means using sample data, which is important in understanding whether a statistically significant difference exists between the two groups.

t-Distribution

The t-distribution is used when the sample sizes are small and the population standard deviations are unknown. It accounts for additional uncertainty by having heavier tails than the normal distribution. This is appropriate for creating confidence intervals for means when working with small samples, ensuring that the interval estimates are accurate despite the limited data.

Standard Error

The standard error measures the variability of a sample statistic, specifically the difference between sample means. It is calculated by combining the variability from both samples and is used to quantify the precision of the difference estimate. In constructing a confidence interval, the standard error is multiplied by a critical value from the t-distribution to determine the margin of error.

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