Question

Do you think that students undergo physiological changes when in potentially stressful situations such as taking a quiz or exam? A sample of statistics students were interrupted in the middle of a quiz and asked to record their pulse rates (beats for a 1-minute period). Ten of the students had also measured their pulse rate while sitting in class listening to a lecture, and these values were matched with their quiz pulse rates. The data appear in Table 4.18 and are stored in QuizPulse10. Note that this is paired data since we have two values, a quiz and a lecture pulse rate, for each student in the sample. The question of interest is whether quiz pulse rates tend to be higher, on average, than lecture pulse rates. (Hint: Since this is paired data, we work with the differences in pulse rate for each student between quiz and lecture. If the differences are \(D=\) quiz pulse rate minus lecture pulse rate, the question of interest is whether \(\mu_{D}\) is greater than zero.) Table 4.18 Quiz and Lecture pulse rates for I0 students $$\begin{array}{lcccccccccc} \text { Student } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \text { Quiz } & 75 & 52 & 52 & 80 & 56 & 90 & 76 & 71 & 70 & 66 \\\ \text { Lecture } & 73 & 53 & 47 & 88 & 55 & 70 & 61 & 75 & 61 & 78 \\\\\hline\end{array}$$ (a) Define the parameter(s) of interest and state the null and alternative hypotheses. (b) Determine an appropriate statistic to measure and compute its value for the original sample. (c) Describe a method to generate randomization samples that is consistent with the null hypothesis and reflects the paired nature of the data. There are several viable methods. You might use shuffled index cards, a coin, or some other randomization procedure. (d) Carry out your procedure to generate one randomization sample and compute the statistic you chose in part (b) for this sample. (e) Is the statistic for your randomization sample more extreme (in the direction of the alternative) than the original sample?

Short answer

The parameters are the mean differences in pulse rates. The null hypothesis is that there is no difference in pulse rates, and the alternative hypothesis is that the quiz pulse rates are higher. The statistic is the mean of the differences, and one way to generate random samples is to shuffle the differences. Compare the test statistic from the original and randomized data to determine whether the original data provides evidence for the alternative hypothesis.

Step-by-step solution

Define Parameters

In this context, the parameter of interest is \(\mu_{D}\), which represents the mean difference in pulse rate for each student between taking a quiz and sitting in lecture.

State the Null and Alternative Hypotheses

Null hypothesis (\(H_{0}\)): \(\mu_{D} = 0\). This means there is no difference, on average, between students’ quiz and lecture pulse rates. \n\nAlternative hypothesis (\(H_{1}\)): \(\mu_{D} > 0\). This implies that the average pulse rate during a quiz is higher than during a lecture.

Determine the Appropriate Statistic

The appropriate test statistic would be the mean value of \(D =\) quiz rate - lecture rate for each student. This is computed by finding the difference between the two rates for each student, adding up these differences, and dividing by the number of students.

Generate Randomization Sample

To generate randomization samples that reflect the paired nature of the data, one could shuffle the differences between the two rates for each student, then assign one difference to each student randomly. Then, these shuffled differences would be used to compute the test statistic.

Compare the Test Statistic Value of the Original and Randomized Samples

After computing the test statistic for one randomization sample in accordance with the chosen method in Step 4, we compare its value to the test statistic calculated for the original sample. If the test statistic from the random sample is more extreme (i.e., farther from zero in the direction of the higher pulse rate during the quiz), it would provide evidence in favor of the alternative hypothesis.