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Statistical significance is an important concept in hypothesis testing that helps us decide if the results of our test are likely due to chance or if they represent true differences. We typically use a significance level, denoted by \(\alpha\), as a threshold. In this exercise, \(\alpha = 0.01\), meaning we are willing to accept a 1% chance of incorrectly rejecting the null hypothesis. This low alpha level is often chosen in contexts where avoiding false positives is critical.

The p-value helps us determine the statistical significance of a test result. It indicates the probability of obtaining a test statistic as extreme as the observed one, assuming the null hypothesis is true. If the p-value is less than \(\alpha\), we say the result is statistically significant, and we reject the null hypothesis. Otherwise, we fail to reject it.

• If the p-value is less than 0.01, the result is statistically significant at \(\alpha = 0.01\).
• If the p-value is greater than 0.01, as in this exercise, it means the data do not provide strong evidence against the null hypothesis.

Statistical significance does not imply that the result is practically important. There can be situations where small, statistically significant differences do not have meaningful real-world implications.

The size of the sample, denoted by \(n\), plays a crucial role in hypothesis testing. It affects both the precision of the parameter estimates and the power of the test. Larger sample sizes typically lead to more reliable results, as they tend to represent the population more accurately.
In the given exercise, the impact of sample size is evident. As \(n\) increases, the test statistic \(z\) computed for the sample averages also changes, reflecting more precise estimates. With larger sample sizes, there is less variability in the sample mean estimate, reducing the standard error \(\sigma/\sqrt{n}\).

• For \(n = 25\), \(z = 0.3125\).
• For \(n = 2500\), \(z = 0.02\).

Larger samples also tend to lower the p-value, making it more likely to find statistical significance if a true effect exists. However, as seen in this problem, a large sample can also highlight tiny differences that aren't practically meaningful, necessitating careful interpretation of results.

The power of a statistical test is the probability that the test correctly rejects a false null hypothesis, i.e., detecting an effect when there is one. It is denoted by \(1 - \beta\), where \(\beta\) is the probability of a Type II error (failing to reject a false null hypothesis).
In the context of this exercise, calculations reveal how power changes with different sample sizes. Given a small true effect, such as \(\mu = 86\) instead of \(\mu = 85\), increasing the sample size increases the power:

• For \(n = 25\), \(\beta = 0.9783\), resulting in very low power.
• For \(n = 2500\), \(\beta = 0.9896\), still indicating low power due to a tiny effect size.

The power of a test is crucial when planning studies. Adequate power is necessary to confidently detect effects. However, as observed, even a test with high power may identify statistically significant results that lack practical importance. This emphasizes the need for balancing statistical and practical significance in interpreting test outcomes.