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Understanding the null and alternative hypotheses is crucial in hypothesis testing. These hypotheses are the backbone of any statistical test and are used to ascertain if there is significant evidence to support a certain belief about a population parameter.

In our example, the null hypothesis (\(H_0\)) is that the mean daily revenue from the vending machine (\(\mu_1\-\mu_2\) did not change after the price increase. Conversely, the alternative hypothesis (\(H_1\) states that the mean daily revenue actually decreased after the price change. This can also be understood as the hypothesis that expects a change or difference, here implying that the mean daily revenue after the price hike (\(\mu_2\) is less than before the hike (\(\mu_1\) In hypothesis testing, we initially assume that the null hypothesis is true and then use statistical evidence to determine whether there is enough data to reject this assumption in favor of the alternative.

Calculating the t-test statistic involves using sample data to measure the difference between the observed sample mean and the proposed population mean, relative to the variability in the sample.

The t-test statistic is calculated using the formula: \[t = \frac{(\bar{x_1} - \bar{x_2}) - D_0}{\frac{s}{\sqrt{n}}}\] This formulation takes into account the sample means (\(\bar{x_1}\) and \(\bar{x_2}\)), the hypothesized difference in means (\(D_0\)), the sample standard deviation (\(s\)) and the sample size (\(n\)). In our exercise, the t-test statistic was used to determine whether the observed sample mean revenue after the price increase significantly differs from the previous mean revenue.

The significance level, denoted as alpha (\(\alpha\)), is a threshold used to determine the probability of committing a Type I error, which occurs when the null hypothesis is incorrectly rejected.

A commonly used \(\alpha\) value is 0.05, meaning there is a 5% chance of rejecting a true null hypothesis. This level of significance is a balance between being too lenient and too strict, allowing researchers to be reasonably confident in their decision to reject or fail to reject the null hypothesis. In the provided exercise, an \(\alpha\) of 0.05 was chosen to test the hypothesis about the average revenue from the vending machine.

Degrees of freedom, often abbreviated as 'df', is a concept closely tied with the estimation of population parameters. It refers to the number of values in a calculation that are free to vary.

In the context of the t-test, the degrees of freedom are calculated as the sample size (\(n\)) minus one (\(df = n - 1\)), which in our example would result in 19 degrees of freedom. This number is essential for determining the correct critical value from the t-distribution table and helps to account for the sample size in the estimation process. The degrees of freedom basically reflect the sample size's contribution to the precision of the estimated population parameter.

The critical value is a point on the distribution of the test statistic beyond which we would reject the null hypothesis. It depends on the chosen significance level and the degrees of freedom.

To determine the critical value, you would usually refer to a t-distribution table or use statistical software. This value serves as a comparison point for our calculated t-test statistic: if the t-test statistic is greater than the critical value, the null hypothesis is rejected. In the example provided, with an \(\alpha\) of 0.05 and 19 degrees of freedom, the critical value was found to be approximately 1.729, which, when compared with our t-test statistic of approximately 5.345, leads to the conclusion that the null hypothesis should be rejected.