Question

Who Exercises More: Males or Females? The dataset StudentSurvey has information from males and females on the number of hours spent exercising in a typical week. Computer output of descriptive statistics for the number of hours spent exercising, broken down by gender, is given: \(\begin{array}{l}\text { Descriptive Statistics: Exercise } \\ \text { Variable } & \text { Gender } & \mathrm{N} & \text { Mean } & \text { StDev } \\\ \text { Exercise } & \mathrm{F} & 168 & 8.110 & 5.199 \\ & \mathrm{M} & 193 & 9.876 & 6.069\end{array}\) \(\begin{array}{rrrrr}\text { Minimum } & \text { Q1 } & \text { Median } & \text { Q3 } & \text { Maximum } \\ 0.000 & 4.000 & 7.000 & 12.000 & 27.000 \\\ 0.000 & 5.000 & 10.000 & 14.000 & 40.000\end{array}\) (a) How many females are in the dataset? How many males? (b) In the sample, which group exercises more, on average? By how much? (c) Use the summary statistics to compute a \(95 \%\) confidence interval for the difference in mean number of hours spent exercising. Be sure to define any parameters you are estimating. (d) Compare the answer from part (c) to the confidence interval given in the following computer output for the same data: Two-sample \(\mathrm{T}\) for Exercise Gender N Mean StDev SE Mean \(\begin{array}{lllll}\mathrm{F} & 168 & 8.11 & 5.20 & 0.40 \\ \mathrm{M} & 193 & 9.88 & 6.07 & 0.44\end{array}\) Difference \(=\operatorname{mu}(F)-\operatorname{mu}(M)\) Estimate for difference: -1.766 \(95 \%\) Cl for difference: (-2.932,-0.599)

Short answer

The dataset includes 168 females and 193 males. On average, males exercise more than females by 1.766 hours. The computed 95% confidence interval for the difference in mean number of hours spent exercising by males and females is (-2.92, -0.61), which is very close to the given confidence interval.

Step-by-step solution

Identifying the Number of Males and Females

From the descriptive statistics, there are 168 females (\(N_F\)) and 193 males (\(N_M\)) in the dataset.

Determining the Group that Exercises More

The mean number of hours spent exercising by females (\(\mu_F\)) is 8.110. The mean number of hours spent exercising by males (\(\mu_M\)) is 9.876. The difference in the mean number of hours spent exercising is \(\mu_M - \mu_F = 9.876 - 8.110 = 1.766\). Hence, on average males exercise more than females by 1.766 hours.

Computing a 95% Confidence Interval

Since we are intending to estimate the difference in means, our parameters would be \(\mu_F\) and \(\mu_M\). The standard errors given for female (\(SE_F\)) is 0.40 and for male (\(SE_M\)) is 0.44. We need to calculate pooled standard error to get the confidence interval: \(SEp = \sqrt{SE_F^2 + SE_M^2} = \sqrt{0.40^2 + 0.44^2} = 0.592\). The 95% confidence interval for the difference in means is computed using the formula: Confidence Interval = Estimate for difference ± \((1.96*SEp)\) = -1.766 ± (1.96 * 0.592)=(-2.92, -0.61).

Comparing with Given Confidence Interval

The confidence interval for the difference computed above is almost similar to the one given in the computer output (-2.932, -0.599). The slight difference might be due to rounding errors.

Key concepts

Confidence Interval

A confidence interval (CI) in statistics is a type of estimate computed from the statistics of the observed data, which gives a range of values that's likely to include the true value of an unknown population parameter. This interval also indicates the degree of uncertainty associated with the estimate of the true difference.

For instance, constructing a 95% CI around a sample mean difference provides us with a range of values within which we can be 95% confident that the true mean difference lies. It's essential to distinguish that this doesn't mean there's a 95% probability the true mean is within this interval; rather, if we were to take many samples and build a CI from each sample, we'd expect about 95% of those intervals to contain the true mean difference.

To improve clarity when working with confidence intervals, it is crucial to:

• Define the level of confidence (e.g., 95%, 99%) and the population parameter being estimated (e.g., mean, proportion).
• Clarify that the interval does not guarantee the parameter lies within it for a given sample.
• Show how the margin of error is computed using the point estimate and the standard error of the mean difference.

In the given exercise, the CI for the mean difference in exercise hours indicates with 95% confidence where the true difference between male and female mean exercise hours lies based on the sample data.

Two-sample T-test

The two-sample T-test is a hypothesis test that allows us to compare the means of two independent groups in order to determine if there is statistical evidence that the associated population means are significantly different. It's extremely useful in studies like the one discussed in our exercise, which compares hours spent exercising between males and females.

When performing a two-sample T-test, assumptions must be met, including that the data is approximately normally distributed and that the variances of the two groups are equal. The test statistic calculated during this test is used to determine if the difference in sample means is significant or if it could have occurred by random chance.

To ensure students fully understand the two-sample T-test, focus on:

• Explaining the null and alternative hypotheses, where the null typically states that there is no difference in population means.
• Discussing the importance of sample size and data distribution.
• Interpreting the significance level (p-value) in the context of rejecting or failing to reject the null hypothesis.

In the exercise provided, the two-sample T-test is used to substantiate whether the observed average difference in exercise hours between genders reflects a true difference in the population or is simply due to sampling variability.

Mean Difference

The mean difference, sometimes referred to as the 'difference in means', 'mean delta', or 'average difference', is a measure used to compare the averages of two distinct groups. In our context, comparing the mean hours spent exercising by males and females reveals the mean difference.

This difference is calculated by subtracting one group's mean from the other's. An important consideration is the variability in the data; even if there is a mean difference, it may not be significant if the data is highly variable.

To effectively communicate about mean differences, educators should:

• Illustrate how to calculate mean difference with clear examples.
• Explain its purpose in understanding group comparisons, such as potential gender differences in exercise habits.
• Discuss the difference's relevance in the real world or practical applications (e.g., program development, policy-making).

In terms of our exercise, the mean difference helps us quantify the average difference in exercise time between males and females, which is the first step in addressing whether a significant difference exists between the two populations.