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The p-value is a crucial concept in hypothesis testing. It helps us decide whether the observed results are statistically significant or happened by random chance. In simpler terms, the p-value tells us the probability of obtaining the test results we did, or more extreme, assuming that the null hypothesis is true.
- A lower p-value indicates that the observed data is unlikely under the null hypothesis, suggesting that the alternative hypothesis might be more plausible.
- In the example above, for each of the cases \(a\), \(b\), and \(c\), the p-value informs us whether the sample mean significantly differs from the hypothesized population mean of 100.

For instance, in Case (b), a p-value of 0.007 indicates a very low probability of obtaining such an extreme test statistic if the null hypothesis were true. This low p-value leads to the rejection of the null hypothesis, suggesting strong evidence against it. Conversely, higher p-values in Cases (a) and (c) suggest insufficient evidence to reject the null hypothesis.

The significance level, denoted by \(\alpha\), is a threshold set by the researcher to determine whether to reject the null hypothesis. It's essentially a criterion for how extreme the data must be to count as strong enough evidence against the null hypothesis.
- Commonly used significance levels are 0.05, 0.01, and 0.10. In the problem above, the significance level is \(\alpha = 0.05\).
- If the p-value is less than \(\alpha\), we reject the null hypothesis, indicating the results are statistically significant.

Consider Case (b) again, where the p-value of 0.007 is less than our significance level of 0.05. This discrepancy provides a strong basis to reject the null hypothesis, implying that the sample mean is likely different from the population mean of 100. However, in Cases (a) and (c), the p-values 0.068 and 0.111 are both greater than 0.05, so there isn't strong enough evidence to reject the null hypothesis based on our chosen level of significance.

The test statistic is a value calculated from the sample data that helps us determine whether to reject the null hypothesis in hypothesis testing. It quantifies the degree of deviation of the sample statistic from the null hypothesis parameter.
- Typically, for means, the test statistic is often a z-score if the population variance is known or based on the sample size and the sample follows normal distribution. The formula used in the provided exercise is:
\[ z = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \]
where \(\bar{x}\) is the sample mean, \(\mu\) is the hypothesized population mean, \(s\) is the sample standard deviation, and \(n\) is the sample size.

In Case \(a\), with \(\bar{x} = 103\), \(s = 11.5\), and \(n = 65\), the calculated z-score is approximately 1.827. This value tells us how many standard deviations the sample mean is from the null hypothesis mean.

The test statistic thus serves as an intermediary step for calculating the p-value, which helps us make the decision: reject or not reject the null hypothesis. Understanding the calculation and interpretation of this value is crucial in evaluating the evidence against the null hypothesis.