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In statistics, the null hypothesis, denoted as \(H_0\), is a statement that there is no effect or no difference. It serves as a default or starting point for hypothesis testing. In the given exercise, the null hypothesis is \(H_{0}: \mu_{1} = \mu_{2}\), meaning that the mean of the first group is equal to the mean of the second group. This hypothesis implies that any observed differences in sample data can be attributed to random chance or sampling variability rather than a true difference in the population means.
Formulating the null hypothesis is crucial because it lays the foundation for the hypothesis testing process. The primary goal of the test is to evaluate whether the data provides sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. It is essential to remember that failing to reject the null hypothesis does not mean it is true, simply that there is not enough evidence to support the alternative.

The two-sample Z-test is a statistical method used to determine whether there is a significant difference between the means of two independent groups. It is applicable when the population variances are known and the sample sizes are reasonably large. In this exercise, a two-sample Z-test is used to compare the means of two groups with known variances, \(\sigma_{1} = 10\) and \(\sigma_{2} = 5\).
To perform the test, we calculate the test statistic 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 \(\bar{x}_{2}\), sample sizes \(n_{1}\) and \(n_{2}\), and variances \(\sigma_{1}^{2}\) and \(\sigma_{2}^{2}\). The computed Z-value helps determine whether to reject the null hypothesis by comparing it to a critical value determined by the significance level \(\alpha\). If the absolute Z-value exceeds the critical value, the null hypothesis is rejected. Otherwise, it is not rejected.

A confidence interval is a range of values, derived from the sample data, that is likely to contain the value of an unknown population parameter. It provides an alternative way to conduct hypothesis testing. In the context of the exercise, a confidence interval for the difference between the two means is constructed as follows:

• \[\bar{x}_{1} - \bar{x}_{2} \pm Z_{\alpha/2} \times \sqrt{\frac{\sigma_{1}^{2}}{n_{1}} + \frac{\sigma_{2}^{2}}{n_{2}}}\]

This formula includes a buffer or margin of error around the observed difference \(\bar{x}_{1} - \bar{x}_{2}\). The critical value \(Z_{\alpha/2}\) corresponds to the desired confidence level, such as 95%, and represents the number of standard deviations needed to capture the central area of the normal distribution.
By constructing this interval, we can see whether it contains the null hypothesized difference of zero. In this case, if the confidence interval includes zero, it suggests there is no statistically significant difference between the means, thus supporting a failure to reject the null hypothesis.

The power of a statistical test, often simply referred to as "test power," is the probability that the test correctly rejects a false null hypothesis. It is essentially the capability of the test to detect a real effect or difference when it truly exists. Mathematically, power is expressed as \(1 - \beta\), where \(\beta\) is the probability of a Type II error (failing to reject a false null hypothesis).
In the exercise, the test power is examined under the condition that \(\mu_{1}\) is 4 units less than \(\mu_{2}\). This scenario assumes an actual difference, allowing us to calculate the test's power by determining the Z-value under the alternative hypothesis and evaluating what portion of the standard normal distribution this Z-value covers.
The power is vital because it influences the likelihood of observing a statistically significant result. Higher power indicates a greater chance of identifying true effects, emphasizing the importance of designing studies with adequate sample sizes to detect meaningful differences.