A quality-control manager at an amusement park feels that...

A quality-control manager at an amusement park feels that the amount of time that people spend waiting in line for the American Eagle roller coaster is too long. To determine if a new loading/unloading procedure is effective in reducing wait time in line, he measures the amount of time (in minutes) people are waiting in line on 7 days. After implementing the new procedure, he again measures the amount of time (in minutes) people are waiting in line on 7 days and obtains the following data. To make a reasonable comparison, he chooses days when the weather conditions are similar. Is the new loading/unloading procedure effective in reducing wait time at the $\alpha=0.05$ level of significance? Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers.

Key Concepts

Paired t-test

A paired t-test is a statistical method used to compare the means of two related groups. In this context, it analyzes the differences in wait times before and after implementing the new loading/unloading procedure, helping determine whether there is a statistically significant change. This test takes into account the paired nature of the data, where each 'before' measurement is directly matched with an 'after' measurement from similar conditions.

Null and Alternative Hypotheses

The null hypothesis typically states that there is no difference in the mean wait times before and after the new procedure, while the alternative hypothesis suggests that the new procedure has led to a reduction in wait times. Formulating these hypotheses clearly is critical for testing whether observed differences can be attributed to the new procedure rather than random variation.

Level of Significance

The level of significance, denoted by alpha (in this case, 0.05), is the threshold used for deciding whether or not to reject the null hypothesis. It represents the probability of committing a Type I error, which is rejecting a true null hypothesis. In practical terms, if the p-value from the test is less than 0.05, it suggests that the observed differences are unlikely due to chance, supporting the effectiveness of the new procedure.

Normality Assumption

For the paired t-test to be valid, the differences between paired observations should be approximately normally distributed. Assessing normality through tools like normal probability plots and boxplots ensures that the statistical test is appropriate and that its conclusions are reliable.

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