1. In all marathons run in the USA during 2020, the average (mean) time for men was 4:25 hours (or 265 minutes) with a standard deviation of 0:58 hours (58 minutes). For women running marathons, the average was 4:52 hours (292 minutes) with a standard deviation of 1:01 hours (61 minutes). Suppose the distributions for marathon times are approximately normal for both men and women. Joe ran the marathon in 5:02 hours (302 minutes), while his wife Marissa finished the marathon in 5:19 hours (319 minutes). Obviously Joe finished the marathon faster than Marissa, but they are curious about how each of them performed according to their genders. Remember: a better performance corresponds to a faster (shorter) finish time.

1. Write down the short-hand for the two normal distributions, for men and for women.

Z = (X-265)/58 = (Y-292)/61

1. What are the Z-scores for Joe's and Marissa's finishing times according to their genders?

(302-265)/58 = 0.638 for Joe

(319-292)/61 = 0.443 for Marissa

1. What do these Z-scores tell you when comparing them?

Joe's time is larger than the mean of male runner times as well as Marissa's compared to the females. Marissa's time is smaller so compared to Joe she is quicker

1. Did Joe or Marissa rank better within their own gender?

Marissa has a smaller Z score so she ranks better within her gender category than Joe does

1. What percent of male marathon runners did Joe finish faster than?

1. What percent of female marathon runners did Marissa finish faster than?

1. If the distributions of finishing times are not nearly normal, would your answers to (b)-(e) be different?

1. According to the above normal distribution for male marathon runners, what would be the cutoff time for the fastest 5% of the men? (that is, those who took the 5% shortest time)

1. According to the above normal distribution for female marathon runners, what would be the cutoff time for the slowest 10% of the women?