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Confidence intervals are a range of values used to estimate a population parameter. In our exercise, we are interested in the difference between the population means of free time for tenth-graders and twelfth-graders. By using a confidence interval, we can express this difference as a range. A 90% confidence interval gives us a 90% assurance that the true difference lies within this interval.

To create the confidence interval, we need the sample means and the standard error of the difference between the sample means. This standard error tells us how much variation we can expect if we were to take multiple samples. The final confidence interval is calculated by taking the difference of the sample means and adding/subtracting a critical value multiplied by the standard error. The critical value is derived from a t-distribution because the population standard deviations are unknown.

Hypothesis testing is a method used to determine whether there is enough evidence to infer that a certain condition is true for a whole population. In this scenario, we want to see if tenth-graders have significantly more free time compared to twelfth-graders.

We start with formulating two hypotheses: the null hypothesis (\(H_0\): \(\mu_1 = \mu_2\)) implies there's no difference between the two groups, while the alternative hypothesis (\(H_1\): \(\mu_1 eq \mu_2\)) suggests a difference exists. By computing a test statistic and comparing its p-value to a pre-determined significance level (5% here), we can decide whether to reject the null hypothesis. If the p-value is less than 0.05, it suggests that it is very unlikely the null hypothesis is true, so we have enough evidence to conclude a difference.

The t-distribution is a probability distribution that we commonly use when dealing with small sample sizes and unknown population standard deviations. Unlike the normal distribution, it has heavier tails, which means it accounts for more variability. This is important when using sample data to estimate population parameters.

In our case, when creating a confidence interval or performing a hypothesis test, we use the t-distribution to find the critical value and the test statistic. The shape of the t-distribution depends on the degrees of freedom, which are calculated based on the sample sizes. Since our samples (25 and 23) are relatively small, using a t-distribution gives us a more accurate picture of variability than the normal distribution.

The sample mean is the average of a set of observations taken from a larger population. It provides an estimate of the population mean, which is often unknown. In our problem, the sample mean for the tenth-graders' free time is 29 hours, while it is 22 hours for the twelfth-graders.

These sample means are central to calculating both the confidence interval and performing hypothesis testing. They act as best estimates for the unknown population means. The difference between these sample means indicates how varied the groups may be. However, we must account for variations in sample estimates using statistical techniques to make accurate inferences about the populations.

Population means represent the average of a characteristic from the entire group we are interested in. Calculating the true population mean can be challenging unless we have access to data from every member of the population. Hence, we rely on sample data to make inferences about population means.

In this context, \(\mu_1\) and \(\mu_2\) are the unknown population means of tenth and twelfth-graders’ free time, respectively. Our goal is to estimate the difference between these two population means. By constructing confidence intervals and running hypothesis tests, we can point out differences between groups in relation to their free time, using sample data as our best guide.