Question

Twelve runners are asked to run a 10-kilometer race on each of 2 consecutive weeks. In one of the races, the runners wear one brand of shoe and in the other, a different brand. The brand of shoe a they wear in each race is determined at random. All runners are timed and are asked to run their best in each race. The results in minutes) are given below.

Use the test for matched pairs to determine if there is evidence that times using Brand 1 tend to be faster than times using Brand 2. What is the approximate) value of the P-value?
A) 0.019
B) 0.042
C) 0.050
D) 0.073

Answer 1

In a matched pairs t-test to determine if Brand 1 shoes result in faster run times than Brand 2, we would calculate the mean differences, standard deviation of the differences, the t-statistic, and correspondingly look up the P-value. Without the actual times, we cannot determine an exact P-value but would reject the null hypothesis if P < 0.05, indicating a significant difference.

The correct answer is B) 0.042

Step-by-step explanation:

To determine if there is evidence that times using Brand 1 tend to be faster than times using Brand 2 in the 10-kilometer race, we would perform a matched pairs t-test. This test compares the two related samples to determine if their means are statistically different. The student has not provided the actual times, so we will focus on the process and consider the P-values provided in the options (A, B, C, D).

The null hypothesis in this case would state that there is no difference in the mean times when using Brand 1 or Brand 2, while the alternative hypothesis would claim that there is a difference, specifically that times with Brand 1 are faster.

We would calculate the differences in each runner's times between the two races, analyze this new data set to obtain the mean difference (bar), the standard deviation of the differences (s_d), and then use the t-statistic formula:

t = (bar - 0) / (s_d/ √n), where 0 is the hypothesized mean difference (typically 0).

After calculating the t-statistic, we look up the corresponding P-value in a t-distribution table using the degrees of freedom (n-1, where n is the number of pairs). If the calculated P-value is less than the significance level (typically α = 0.05), we reject the null hypothesis, suggesting a significant difference in times.

The correct answer is B) 0.042