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The t-test is a type of hypothesis test used when the population standard deviation is unknown and the sample size is relatively small. It helps determine whether there is a significant difference between a sample mean and a known value (often a population mean). There are different types of t-tests, including:

• One-sample t-test: Compares the sample mean to a known value or theoretical expectation.
• Independent t-test: Compares the means of two independent groups.
• Paired t-test: Compares means from the same group at different times or under two different conditions.

In our exercise, the t-test is needed because the population standard deviation, denoted by \( \sigma \), is unknown for parts a and c. The formula for the t-test statistic is:\[t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}\]where \( \bar{x} \) is the sample mean, \( \mu_0 \) is the population mean under the null hypothesis, \( s \) is the sample standard deviation, and \( n \) is the sample size. The goal is to compare this value against a critical value to decide whether to reject the null hypothesis.

The z-test is another method of hypothesis testing, typically used when the population standard deviation is known and the sample size is larger. It assesses if a sample mean significantly differs from a hypothesized population mean. The types are similar to those of the t-tests, such as one-sample z-test and two-sample z-test.In part b of our exercise, the z-test is appropriate because the population standard deviation, \( \sigma \), is given. The z-test statistic formula is:\[z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}\]where \( \bar{x} \) is the sample mean, \( \mu_0 \) is the population mean under the null hypothesis, \( \sigma \) is the population standard deviation, and \( n \) is the sample size. The calculated z-value is then compared to a critical z-value obtained from the standard normal distribution to make a decision regarding the null hypothesis.

The critical value is a point on the test statistic distribution that is compared against the calculated test statistic to decide whether to reject the null hypothesis. It depends on the significance level \( \alpha \) and the nature of the test (one-tailed or two-tailed).

• One-tailed test: Uses one critical value in one direction, either left or right, based on the hypothesis.
• Two-tailed test: Has two critical values, one on each tail, allowing for deviations in both directions from the hypothesized mean.

For the t-tests in parts a and c of our exercise, critical t-values are used, which depend on the degrees of freedom (\( df = n - 1 \)). In part a, the critical t-value was approximately \( -1.304 \) for a left-tailed test at \( \alpha = 0.10 \). For the z-test in part b, the critical z-value was \( 1.645 \) for a right-tailed test at \( \alpha = 0.05 \). These critical values determine the threshold for rejecting the null hypothesis.

The test statistic is a standardized value computed from sample data during a hypothesis test. It quantifies the degree of deviation from the null hypothesis. The kind of test statistic (t or z) depends on what information is known about the population.

• For a t-test, the test statistic is calculated using the sample standard deviation \( s \) (when \( \sigma \), the population standard deviation, is unknown).
• For a z-test, the population standard deviation \( \sigma \) is known, and thus, the test statistic is computed with it.

The test statistic aggregates sample data in a way that can be directly compared to a critical value from a statistical distribution. For example, in part a of our exercise, the t-test statistic was approximately \( -2.18 \). By contrast, in part b, the z-test statistic was approximately \( 1.61 \). These values help in determining if the results are statistically significant enough to reject the null hypothesis.