Waiting in Line A quality-control manager at an amusement...

Waiting in Line 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. (TABLE CANNOT COPY)

Key Concepts

Null and Alternative Hypotheses

In a hypothesis testing framework, the null hypothesis represents the default position that there is no effect or difference—in this case, that there is no reduction in wait time. The alternative hypothesis represents the claim that there is an effect—in this context, that the wait time has been reduced due to the new procedure.

Level of Significance

The level of significance, denoted by ?, is the threshold for rejecting the null hypothesis. It defines the probability of making a Type I error, which is rejecting the null hypothesis when it is true. An ? of 0.05 means that there is a 5% risk of concluding that the procedure is effective when, in fact, it is not.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. In this context, it means setting up a null hypothesis (that the new procedure does not reduce the wait time) and an alternative hypothesis (that the new procedure reduces the wait time), and then using sample data to test these hypotheses.

Paired t-Test

The paired t-test is used when the data consist of paired or matched observations, such as measurements taken from the same subjects under different conditions. It assesses whether the mean difference between the paired observations is statistically significantly different from zero, making it an appropriate test when comparing waiting times before and after a procedural change.

Assumption of Normality

The assumption of normality means that the differences between paired observations are assumed to be normally distributed. This is an important condition for the validity of the paired t-test, as it allows the use of the t-distribution to perform the hypothesis test, particularly in smaller samples.

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