Kathy and her brother Clay recently ran in a local marathon. The distribution of finishing time for women was approximately norm

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Kathy and her brother Clay recently ran in a local marathon. The distribution of finishing time for women was approximately norm

al with mean 259 minutes and standard deviation 32 minutes. The distribution of finishing time for men was approximately normal with mean 242 minutes and standard deviation 29 minutes. (a) The finishing time for Clay was 289 minutes. Calculate and interpret the standardized score for Clay's marathon time. Show your work. (b) The finishing time for Kathy was 272 minutes. What proportion of women who ran the marathon had a finishing time less than Kathy's? Show your work. (c) The standard deviation of finishing time is greater for women than for men. What does this indicate about the finishing times of the women who ran the marathon compared to the finishing times of the men who ran the marathon?

Mathematics

1 answer:

Inessa [12.5K] · Answer 1 · 2026-02-15T21:09:45+03:00

Part a)

The finishing times for men followed an approximately normal distribution with a mean of 242 minutes and a standard deviation of 29 minutes.

We need to evaluate and interpret the standardized score for Clay's marathon time of 289 minutes.

The relevant formula is:

$z = \frac{x - \bar x}{s}$

Inserting the values yields:

$z = \frac{289 - 242}{29}$

$z = 1.62$

This indicates that Clay's marathon time is 1.62 standard deviations higher than the average finishing time.

Part b)

In this instance, the women's finishing times also displayed an approximate normal distribution, with a mean of 259 minutes and a standard deviation of 32 minutes.

We aim to determine the percentage of women completing the marathon in a time shorter than Kathy's, who finished in 272 minutes.

We first need to calculate the z-score:

$z = \frac{272 - 259}{32} = 0.41$

According to the standard normal distribution table, P(z<0.41)=0.6591.

This indicates that 65.91% of female participants had a finishing time less than that of Kathy.

Part c

The standard deviation measures the extent of deviation of individual data points from the mean.

A higher standard deviation for women’s finishing times compared to men suggests that women’s finishing times are more spread out from the average than those for men.