Question

Not all marathons are the same. Some have challenging hills, weather, and competition. Others are known for being a "fast" course or "downhill." Finishing any marathon is a challenge. Let's compare two marathons from recent years using z-scores and the normal distribution.  Data from marathonguide.com.

Marathon #1: Mt. Lemmon, AZ	Marathon #2: Los Angeles, CA
mean finish time: 4.344.34 hours standard deviation: 0.850.85 hour N=421N=421	mean finish time: 5.3155.315 hours standard deviation: 1.18031.1803hour N=21626N=21626

1. What can we conclude from the means of the two races?

The average finisher in the Mt. Lemmon, AZ marathon is Select an answer slower than faster than about the same as Correct the average finisher in Los Angeles, CA marathon because the mean finishing time for Mt. Lemmon, AZ is Select an answer greater than less than about equal to Correct Los Angeles, CA.

2. What can we conclude from the standard deviations of the two races?

The finishing times in the Mt. Lemmon, AZ marathon are Select an answer further apart closer together not different Correct compared to the Los Angeles, CA marathon because the standard deviation in Mt. Lemmon, AZ is Select an answer greater than less than equal to Correct Los Angeles, CA.

3. Use z-scores (rounded to 2 decimal places) and the normal distribution to determine what percentage of finishers in each marathon finished with the following:

Marathon #1: Mt. Lemmon, AZ	Marathon #2: Los Angeles, CA
Under 3 hours: z score =   Percent < 3 hours:     %	Under 3 hours: z score =    Percent < 3 hours:     %
Over 5.25 hours: z score =   Percent > 5.25 hours:    %	Over 5.25 hours: z score =    Percent > 5.25 hours:    %
Between 3 and 5.25 hours:     %	Between 3 and 5.25 hours:     %

4. Based on your analysis, what can you conclude about the difficulty of each marathon? Explain your thinking referring to the data.

Answer 1

1. The average finisher in the Mt. Lemmon, AZ marathon is faster than the average finisher in Los Angeles, CA marathon because the mean finishing time for Mt. Lemmon, AZ is less than Los Angeles, CA.

2. The finishing times in the Mt. Lemmon, AZ marathon are closer together compared to the Los Angeles, CA marathon because the standard deviation in Mt. Lemmon, AZ is less than Los Angeles, CA.

3.

Following formula can be used to calculate the z score

Mount Lemmon:

Under 3 hours:

The p value is 0.05699

Over 5.25 hours:

the p value is 0.14472

Percentange between 3 and 5.25 hours is 1 - (0.14472 + 0.05699) = 0.79829

Los Angeles:

Under 3 hours:

the p value = 0.02499

Over 5.25 hours:

THe p value is 0.478

The percentage between 3 and 5.25 hours is 1 - 0.478 - 0.02499 = 0.49701

4. Based on the analysis, it seems that Los Angeles Marathon is more difficult when compared with Mt. Lemmon because it is clear the more percentage of people are able to finish the marathon between 3 and 5.25 hours when compared with Los Angeles.