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When you have a small sample size and want to compare the sample mean to a known value, a t-test helps you determine if there's a significant difference or not. In this exercise, the null hypothesis (H_0: \mu = 22.5) tests if the average interior temperature is 22.5°C. The alternative hypothesis (H_1: \mu eq 22.5) suggests it's different. To assess these hypotheses, we calculate the t-statistic. It's done using the formula: \[ t = \frac{\overline{x} - \mu}{s / \sqrt{n}} \] where \( \overline{x} \) is the sample mean, \( \mu \) is the hypothesized mean, \( s \) is the sample standard deviation, and \( n \) is the sample size.

Check the calculated t-value against the critical t-value from statistical tables at your chosen significance level (\( \alpha \)). Here, \( \alpha = 0.05 \), meaning you're allowing a 5% chance of rejecting the null hypothesis when it's actually true (type I error). If the t-value is beyond this critical range, it indicates a statistically significant difference, leading you to reject \( H_0 \).

Additionally, the p-value shows the probability of observing the test results under the null hypothesis. A p-value less than \( \alpha \) indicates strong evidence against \( H_0 \).

Before performing a t-test, it's crucial to ensure your data is approximately normally distributed. This is because the t-test relies on the normality assumption for small sample sizes (typically \( n < 30 \)).

The Shapiro-Wilk test is a common tool to assess normality. It provides a p-value to gauge whether your data deviates from a normal distribution. If this p-value is greater than 0.05, you don't have enough evidence to say your data isn't normal, thus, you can proceed with the t-test safely.

Alternatively, graphical methods like Q-Q plots or histograms also help visualize data normality. A Q-Q plot compares your data quantiles against a normal distribution. If points lie along the line, your data is likely normal. Histograms offer a visual check – a bell-shaped curve suggests normality.

The normality check is essential because if your data is far from normal, your t-test results might not be valid, leading to incorrect conclusions.

To further validate hypothesis testing, constructing a confidence interval (CI) offers another perspective. A 95% CI gives a range within which we expect the true mean to fall 95% of the time. In our example, the formula is: \[ \overline{x} \pm t_{\alpha/2} \times \left( \frac{s}{\sqrt{n}} \right) \] where \( t_{\alpha/2} \) is the t-value for \( 1 - \alpha/2 \) confidence, \( \overline{x} \) is the sample mean, \( s \) is the standard deviation, and \( n \) is the sample size.

If the hypothesized mean (22.5) lies outside this interval, it provides evidence against the null hypothesis, aligning with p-value results. Thus, CIs offer a confidence range rather than a simple yes/no hypothesis test result, crowning it a valuable tool for deeper insight.

Also, wider CIs suggest more variability in data while narrower CIs imply more precise estimates of the mean. So, confidence intervals not only support decision-making in hypothesis tests but also reveal estimation reliability.

Determining the needed sample size for sufficient statistical power is an important step. If you want the power of a test to be at least 0.9, it ensures a 90% probability of correctly rejecting a false null hypothesis (type II error).

To compute required sample size \( n \), use this equation: \[ z_{\alpha/2} + z_{power} = \frac{\delta}{s / \sqrt{n}} \] where \( z_{\alpha/2} \) and \( z_{power} \) are standard normal z-values corresponding to the desired significance level and power.

The non-centrality parameter \( \delta \) relates to the difference you'd like to detect (true mean - hypothesized mean). Solving for \( n \) gives you the minimum sample size needed, ensuring your test's reliability.

Appropriate sample size benefits testing accuracy, saving resources, and avoiding unnecessary sampling. It leads to more reliable and confident conclusions and prevents weak statistical power that could miss a true effect.