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In hypothesis testing, the test statistic is a crucial element. It helps determine whether to reject the null hypothesis. When comparing two means with known variances, the test statistic is often calculated using the formula:\[Z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}\]This formula combines the sample means (\(\bar{x}_1\) and \(\bar{x}_2\)), the known variances (\(\sigma_1\) and \(\sigma_2\)), and the sizes of the samples (\(n_1\) and \(n_2\)). By substituting the given values, we determine if there's enough statistical evidence to support our alternate hypothesis.The result tells us how many standard deviations our observed difference is from the expected difference under the null hypothesis. This guides us in making decisions regarding our hypotheses.

The p-value is a probability measure that helps us interpret the result of a hypothesis test. It quantifies the evidence against the null hypothesis.- A small p-value (typically \(\leq\) 0.05) indicates strong evidence against the null hypothesis, suggesting it is unlikely to be true.- A large p-value (\(\geq\) 0.05) suggests weak evidence against the null hypothesis, meaning we fail to reject it.In our test, with a p-value of approximately 0.1736, this indicates there's not enough evidence to reject the null hypothesis using an alpha level of 0.01. We interpret this to mean that there isn't significant evidence to conclude that the first population mean is greater than the second.

Confidence intervals provide a range of values within which we believe the true parameter lies, with a certain level of confidence.To create a confidence interval for the difference between two means, we use:\[(\bar{x}_1 - \bar{x}_2) \pm Z_{\alpha/2} \times \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}\]In this case, for a 99% confidence level, we find an interval of \([-5.71, 12.71]\). Since this interval includes zero, it implies no significant difference between the means at the 1% level. Confidence intervals give us a practical way of considering hypothesis testing, offering a 'range' rather than a simple "accept/reject" decision.

Calculating an appropriate sample size is essential for ensuring our test has sufficient power to detect a true effect.To compute the sample size based on desired power and type I and II errors, we use:\[n = \left( \frac{Z_{\alpha} + Z_{1-\beta}}{\delta / \sqrt{\frac{\sigma_1^2 + \sigma_2^2}{2}}} \right)^2\]This formula considers the effect size \(\delta\), the critical values for \(\alpha\) (type I error) and \(\beta\) (type II error). It ensures a balance between finding true effects and minimizing false negatives or positives.For our scenario, to achieve a power of 0.95 (\(\beta = 0.05\)), the required sample size per group is about 44. Such calculations help design effective experiments that address the research questions properly with adequate power to detect true differences when they exist.