Recall Example 6.2 about the bike commuting times. On each of 56 days (from mid-January to mid-July 2010), researcher Jeremy Groves tossed a £1 coin to decide whether he would be biking the 27 miles to work on his carbon frame (lighter) bicycle that weighed 20.90 lb or on his steel frame (heavier) bicycle that weighed 29.75 lb. He then recorded the commute time (given in minutes) for each trip. The summary statistics were reported to be:

In Example 6.2 you read about conducting a randomization based test to investigate whether the average commute time differs by bike type.
a. Suppose we suspect that the lighter carbon frame bike will be less stable, especially in inclement weather conditions, and will have a higher variability in commute times, compared to the heavier steel frame bike. Is there any indication from the data in support of this suspicion? Explain how you are deciding. To investigate whether the variability in commute times will be higher with the lighter carbon frame bike than the heavier steel frame one, we can examine the ratio of the SDs, such that Ratio = SD_carbon Ã· S_Dsteel. (Note: ÷ is €œdivided by.€)
b. In theory, what numeric values can this ratio take?
c. For the observed data, what is the numeric value of the statistic Ratio?
d. Describe in detail how you would use simulation (with pennies, dice, spinners, cards, etc.) to find a p-value to test whether the variability in commute times is higher with the lighter carbon frame bike than the heavier steel frame one. Here is a histogram from an appropriately conducted simulation:

e. As per the above histogram, what numeric values are most common for Ratio from the simulation? Why does that make sense?
f. Indicate, on the above histogram, the region that depicts the p-value in the context of the study, being sure to include appropriate numbers.
g. Which of the following appears to be the approximate p-value? No explanations needed. Choose one.

A. 1.00 

B. 0.10
C. 0.01 

D. < 0.01

h. Which of the following is an appropriate statement to make based on the above histogram? Circle one.
A. There is not convincing evidence of the variability in commute times being higher with the lighter carbon frame bike than the heavier steel frame one because the distribution of Ratio is not centered at 0.
B. There is some evidence of the variability in commute times being higher with the lighter carbon frame bike than the heavier steel frame one because the distribution of Ratio is centered not at 0 but at 1.
C. There is not convincing evidence of the variability in commute times being higher with the lighter carbon frame bike than the heavier steel frame one because the distribution of Ratio is not normal but skewed right.
D. None of the above.

Transcribed Image Text:

Sample Sample average Sample Bike type Carbon frame (lighter) Steel frame (heavier) size SD 108.34 min 6.25 min 26 107.81 min 4.89 min 30 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Ratio