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When comparing two population means, the point estimate helps to identify the estimated difference between them based on sample data. It is the single best guess of the actual value. In this exercise, we are given two independent random samples with the following details:

• Sample 1 has a mean of 22.5
• Sample 2 has a mean of 20.1

The point estimate for the difference between these two population means is found by subtracting the mean of Sample 2 from the mean of Sample 1. To put it simply, it is calculated as:\[\bar{x}_1 - \bar{x}_2 = 22.5 - 20.1 = 2.4\]This means that, based on the sample data, it is estimated that Sample 1 has a mean that is 2.4 units greater than Sample 2. Point estimates, like this difference, provide a starting point for further statistical analysis, such as constructing a confidence interval.

Degrees of freedom (df) are a crucial aspect of many statistical calculations, including the two-sample t-test used here. They represent the number of independent values or quantities which can vary freely. For a two-sample t-test, the degrees of freedom help in determining the critical value from the t-distribution, which is used to construct confidence intervals.In this exercise, we utilize the Welch-Satterthwaite equation to calculate the degrees of freedom for two independent samples. It takes into account the sample variances and sizes:\[df = \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{\left( \frac{s_1^2}{n_1} \right)^2}{n_1 - 1} + \frac{\left( \frac{s_2^2}{n_2} \right)^2}{n_2 - 1}}\]After performing the detailed calculations, we find the degrees of freedom to be approximately 46. This is a rounded value and is essential for accurately determining the t critical value needed for the margin of error and confidence interval calculations.

The margin of error provides an indication of the potential error in the point estimate. It defines the range in which we expect our population parameter to lie, with a certain level of confidence, typically 95% or 99%.To calculate the margin of error for the difference between two population means, we use the critical value from the t-distribution and the standard error of the difference. The critical value, denoted as \(t^*\), is determined based on the confidence level and degrees of freedom. Here, with df = 46 and a 95% confidence level, \(t^* \approx 2.013\).The standard error (SE) is calculated using:\[\text{SE} = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\]For our samples, this comes out to approximately 1.0399.The margin of error is then obtained by multiplying \(t^*\) with the SE:\[\text{Margin of Error} = t^* \times \text{SE} = 2.013 \times 1.0399 \approx 2.093\]This means we can be 95% confident that the true difference between the population means lies within 2.093 units of our point estimate, providing a clearer picture of the variability present in the estimate. This step aids in making informed decisions based on the data.