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Pooled variance is a method used to estimate the variance of two populations that are assumed to have the same variance. In a two-sample t-test, it's crucial to combine the variances of both samples into a single, pooled estimate. This pooled variance represents the weighted average of the variances from both groups.

To calculate it, we use the formula \[ (n_A - 1)s_A^2 + (n_B - 1)s_B^2 \/ (n_A + n_B - 2) \], where \(n_A\) and \(n_B\) are the sample sizes and \(s_A^2\) and \(s_B^2\) are the sample variances. The degrees of freedom for each sample, \(n_A - 1\) and \(n_B - 1\), ensure that the variance is appropriately weighted for each group's contribution.

For the given problem, substituting the numbers into the formula yielded a pooled variance of 254.22. This step is foundational, as the pooled variance is used in computing the t-statistic, which ultimately helps us determine if there's a significant difference between the groups' means.

Next comes the t-statistic calculation, which is the centerpiece of the two-sample t-test. The t-statistic reflects how much the sample means differ from each other relative to the variation within the samples. To compute the t-statistic, we apply the formula \[ (\bar{X}_A - \bar{X}_B) \/ \sqrt{(PooledVariance/n_A) + (PooledVariance/n_B)} \], where \(\bar{X}_A\) and \(\bar{X}_B\) are the sample means.

What's happening here is we're scaling the difference between the sample means by the standard error, which spreads the variance across the combined sample sizes. This calculation tells us the number of standard errors away the sample means are from one another. With the values given, the t-statistic turned out to be 1.26, a measure that we will use to assess the significance of the difference observed.

Degrees of freedom is a concept tied to the reliability of a statistical estimate. It's roughly the number of independent pieces of information that go into the estimate. For a two-sample t-test, the total degrees of freedom are calculated by adding the sample sizes of both groups, then subtracting two (the number of sample means estimated).

The formula is \(df = n_A + n_B - 2\). Degrees of freedom are crucial when referring to the t-distribution, as they shape the distribution. More degrees of freedom typically mean the distribution is closer to the normal distribution. In our exercise, with 8 participants in group A and 15 in group B, we have a total of 21 degrees of freedom. This figure is intrinsic to the final steps of hypothesis testing, where we compare our t-statistic against critical values from the t-distribution.

Finally, hypothesis testing is a systematic method to decide whether to accept or reject a hypothesis about a population parameter, based on sample data. In this case, we're testing the null hypothesis \(H_0: \mu_A = \mu_B\) against the alternative hypothesis \(H_a: \mu_A eq \mu_B\).

We compare the calculated t-statistic to a critical value from the t-distribution that corresponds to the degrees of freedom from our test and the desired level of significance. If our t-statistic exceeds the critical value, we reject the null hypothesis, indicating a statistically significant difference between the groups.

Using a t-table or software, we find the critical value for a two-tailed test with 21 degrees of freedom at a 5% significance level is approximately 2.08. Since our calculated t-statistic of 1.26 is less than this critical value, we do not reject the null hypothesis. Therefore, there isn't sufficient evidence to conclude that there is a significant difference between the means of Group A and Group B.