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A confidence interval is a range of values, derived from the sample data, that is likely to contain the population parameter of interest. When we calculate, for example, a 95% confidence interval, we are saying that we are 95% confident that the true population mean falls within this range.

In the context of the exercise, the confidence intervals are given for the population mean \( \mu \). When the sample mean falls within the confidence interval, such as in parts (a) and (c), we do not have sufficient evidence to reject the null hypothesis. This is because the hypothesized value of \( \mu=15 \) is within the range that we're 95% or 90% confident contains the true mean.

However, for part (b), the hypothesized mean does not fall within the 95% confidence interval. This suggests that the true mean is likely different from 15, and we would reject the null hypothesis in this scenario. Confidence intervals are an essential part of statistical hypothesis testing because they provide a range of plausible values for the parameter, allowing for a decision on the null hypothesis based on the data at hand.

The null hypothesis, denoted as \( H_0 \), is a statement that there is no effect or no difference, and it generally represents a skeptical perspective or a claim to be tested. In hypothesis testing, we seek to determine whether the evidence suggests that we should reject this null hypothesis in favor of an alternative hypothesis, denoted as \( H_a \).

For instance, in the given exercise, \( H_0: \mu=15 \) asserts that the population mean is 15. The alternative hypothesis \( H_a: \mu eq 15 \) posits that the population mean is not 15. The null hypothesis is the starting assumption for the test, and the hypothesis testing procedure examines whether the data collected provides enough evidence to conclude if the null hypothesis can be rejected or not.

As we can see from the solutions, a confidence interval that does not include the value stated in the null hypothesis (as in part (b)) is an indicator that the null hypothesis may not hold. Conversely, when the confidence interval includes the null hypothesis value, we lack evidence to reject it (as seen in parts (a) and (c)).

The significance level, often denoted by \( \alpha \), is the probability of rejecting the null hypothesis when it is actually true, known as a Type I error. It represents the researcher's tolerance for such errors and is a critical value in hypothesis testing that helps determine the threshold for rejecting the null hypothesis.

Common significance levels are 5% (0.05), 1% (0.01), or 10% (0.10), which corresponds inversely to 95%, 99%, and 90% confidence levels, respectively. Demonstrated in our exercise, when we reject the null hypothesis for part (b), it's because the 95% confidence interval does not include the hypothesized mean of 15, thus surpassing the 5% significance level criterion for rejection.

Alternatively, for parts (a) and (c), the hypothesized mean lies within the confidence intervals, indicating that we do not have significant evidence at the 5% and 10% levels, respectively, to reject the null hypothesis. Deciding on the appropriate significance level is a crucial step in the design of an experiment or study as it can influence the conclusions drawn from the statistical test.