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The point estimate provides a single value as an estimate of an unknown population parameter. In the context of comparing two populations, such as samples from different groups, the point estimate refers to the difference between the sample means.
This difference represents an estimate of how much one population mean deviates from the other.
In our given exercise, we calculate the point estimate by simply subtracting the mean of Sample 2 from the mean of Sample 1:

• Sample 1 mean = 22.5
• Sample 2 mean = 20.1
• Point estimate = 22.5 - 20.1 = 2.4

Thus, the point estimate of the difference between the two population means is 2.4.
This number suggests that, based on our samples, the mean of the first population is 2.4 units higher than that of the second population.

Degrees of freedom (df) are an integral component in statistical analysis, especially when dealing with variance and mean estimates. They are essentially the number of independent values that can vary in an analysis without breaking any constraints established on the data.
In our case, while comparing two samples with possibly different variances, Welch's approximation calculates the degrees of freedom. This formula gives a more precise account when the sample variances differ.
Welch's approximation uses the formula:\[df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}}\]Here:

• \(s_1 = 2.5\), \(s_2 = 4.8\)
• \(n_1 = 20\), \(n_2 = 30\)

and estimates the degrees of freedom to be approximately 190 after computation, helping refine the hypothesis test results.

Welch's approximation specifically addresses challenges when working with two independent samples having unequal variances. This method adjusts the degrees of freedom calculation in a two-sample t-test.
By not assuming equal population variances, Welch's method increases accuracy in estimating test statistics for more reliable outcomes.
In this exercise, we utilize the formula to compute degrees of freedom when sample variances are not equal, providing flexibility in statistical tests:

• It's especially useful when sample sizes and variances differ significantly.
• It avoids underestimating the variability and thus maintains the integrity of the test results.

The calculated degrees of freedom using Welch's approximation was approximately 190, ensuring that the confidence interval estimation is both accurate and robust.

The margin of error in statistics represents the extent of potential variation in a sample estimate, offering a range that likely contains the true population parameter.
It is especially crucial in confidence intervals, as it determines the range either side of a point estimate.
To calculate the margin of error (E) for the difference between two means, we use:\[E = t^* \times \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \]Here:

• The critical t-value \(t^*\) for \(df = 190\) is approximately 1.972.
• Substituting the variances and sample sizes, we find \(E = 2.05\).

This value indicates that the true difference in means is expected to be within 2.05 units of our point estimate, with a 95% confidence.