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A Z-test is a type of statistical test used to determine whether there are meaningful differences between two population means. Considered a parametric test, it requires that certain conditions, like normal distribution of the data and known variances, are met.
For this exercise, we're dealing with two sample means: 24.5 and 21.3, with known variances. The calculation of the Z-test statistic involves computing the difference between sample means and dividing by the standard error of the difference.

• This critical value allows us to determine whether the observed differences between means are statistically significant, at a given level of significance (in the example, 0.01).

The Z-test is emphasizing normal distribution and knowledge of variance, which makes it specific, albeit powerful, under the right conditions.

Confidence intervals provide a range of values which are believed to contain the true difference between two population means. They give us an intuitive sense of how much the sample means could vary if the experiment were repeated.
In our exercise, the confidence interval for the difference is calculated as 3.2, adjusted by the critical value times the standard error. This forms the boundaries -

• Lower Boundary: 3.2 − 7.99 = -4.79
• Upper Boundary: 3.2 + 7.99 = 11.19

The interval [-4.79, 11.19] encompasses zero, which suggests that there isn't enough evidence to indicate a significant difference at our 1% significance level. Confidence intervals provide additional insights beyond hypothesis testing, helping substantiate or challenge conclusions drawn from statistical tests.

The P-value indicates the probability of observing results as extreme as, or more so than, the actual results (*assuming the null hypothesis is true*). It determines how surprised we'd be if the test result were observed again.
For a Z-test with a statistic of 0.935, we compute the P-value using the standard normal distribution table.

• P-value = 1 - 0.8243 = 0.1757

Because our P-value (0.1757) exceeds the significance threshold (0.01), we lack sufficient evidence to reject the null hypothesis. The P-value serves as a crucial indicator, providing a clear measure of statistical significance versus arbitrary cutoffs.

The power of a test reflects the likelihood of correctly rejecting a false null hypothesis. It indicates how well the test can detect effects when they genuinely exist. High power is always desirable as it implies greater reliability of test results.In our example, if the true difference in means (\( u \)) is 2, we calculate the power by first computing the non-centrality parameter and then determining the probability of obtaining a relevant Z-score:

• Non-centrality parameter *\( \delta \)* = *\( 0.58 \)*
• Power is the probability of *Z* > *1.75*, approximately *0.0401*.

Our calculated power here indicates a low probability of detecting a true difference, suggesting improvement or adjustments might be needed for better test reliability.

Determining the proper sample size is vital, influencing the balance between statistical power and practical limitations. It ensures that tests carried out are both efficient and accurate.For equal sample sizes with desired power (1-\( \beta \)) of 0.05:

• Use *\( Z_{\alpha} \approx 1.645 \)* and *\( Z_{\beta} \approx 1.645 \)*
• Calculate using: \[ n = \left( \frac{(Z_{\alpha} + Z_{\beta}) \times \sqrt{2 \times 5}}{2} \right)^2 \]
• Result is approximately 32 samples per group.

An appropriately sized sample enables more effective testing, offering stronger confidence in results and minimizing risks of false conclusions due to sampling variability.