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Hypothesis testing is a statistical method used to make decisions or inferences about a population based on sample data. It typically involves two hypotheses: the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)). The null hypothesis is a statement of no effect or no difference, and it's what we assume to be true unless there's enough evidence to suggest otherwise. In this exercise, the null hypothesis is \(H_0: \mu = 100\)—meaning we expect the population mean to be 100.

The alternative hypothesis is what we're trying to gather evidence for; it suggests a change or difference. Here, the alternative hypothesis is \(H_a: \mu < 100\), indicating we're testing if the population mean is less than 100. We use statistical tests, like the t-test, to determine if the sample data provides sufficient evidence to reject the null hypothesis in favor of the alternative.

The t-statistic is a value that results from a statistical hypothesis test. It allows us to determine how far our sample mean is from the null hypothesis population mean, in terms of standard errors. For a hypothesis test of a mean, the t-statistic is calculated using the formula: \[ t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \]

In this equation:

• \(\bar{x}\) is the sample mean.
• \(\mu\) is the hypothesized population mean under the null hypothesis.
• \(s\) is the sample standard deviation.
• \(n\) is the sample size.

The magnitude of the t-statistic helps us understand how likely it is to obtain a sample mean this extreme if the null hypothesis is true. In our case, a t-statistic of \(-2\) indicates that the sample mean is 2 standard errors below the null hypothesis mean.

The sample mean, denoted as \(\bar{x}\), is the average value of a sample, calculated by adding up all the individual observations and dividing by the total number of observations. In statistics, the sample mean is crucial because it serves as an estimator of the population mean—it's our best guess in trying to understand the larger population from the sample data we have.

The sample mean in our exercise is \(98\). It suggests that, based on the 16 sampled observations, the average is less than the hypothesized population mean of 100. The difference between the sample mean and the population mean of the null hypothesis (\(\bar{x} - \mu\)) is a key component in computing the t-statistic and ultimately deciding if we should reject or fail to reject the null hypothesis.

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve. It is symmetric around the mean, which makes it a very important distribution in statistics. Many statistical tests, including the t-test, are based on the assumption that the sample mean is from a normally distributed population, especially with small sample sizes.

In the given exercise, we assume our sample is drawn from a normal population, meaning the properties of the normal distribution apply. These properties help justify using the t-test for hypothesis testing in such situations. The normal distribution allows us to apply the central limit theorem, which says that the distribution of the sample mean approaches a normal distribution as the sample size becomes large—even if the original population is not normally distributed. However, since we have a small sample size of 16 here, the assumption of a normal distribution in the population is especially useful for ensuring our hypothesis test is valid.