Question

1. show all your work. indicate clearly the methods you use, because you will be scored on the correctness of your methods as on the accuracy and completeness of your results and explanations.

kathy and her brother clay recently ran in a local marathon. the distribution of finishing time for women was approximately normal 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 finishing 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 greater 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 variability of the finishing times of the women who ran the marathon compared to the finishing times of the men who ran the marathon?
(b) summary statistics for the sizes are given in the following table.

mean	standard deviation	min	q1	median	q3	max

determine whether there are potential outliers in the data. then use the following grid to sketch a box - plot of room size.
(c) what characteristic of the shape of the distribution of room size is apparent from the histogram but not from the box - plot?

Explanation:

Step1: Calculate Clay's standardized score (z - score)

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x$ is the data - point, $\mu$ is the mean, and $\sigma$ is the standard deviation. For men, $\mu = 242$ and $\sigma=29$, and $x = 289$.
$z=\frac{289 - 242}{29}=\frac{47}{29}\approx1.62$
Interpretation: Clay's finishing time is approximately 1.62 standard deviations above the mean finishing time for men.

Step2: Calculate the proportion of women with a finishing time less than Kathy's

First, find the z - score for Kathy. For women, $\mu = 259$ and $\sigma = 32$, and $x = 272$.
$z=\frac{272-259}{32}=\frac{13}{32}\approx0.41$
Using a standard normal table (z - table), the proportion of values to the left of $z = 0.41$ is approximately $0.6591$. So the proportion of women who had a finishing time less than Kathy's is about $0.6591$.

Step3: Interpret the standard deviation difference

A larger standard deviation for women ($\sigma_{women}=32$) compared to men ($\sigma_{men}=29$) indicates that the finishing times of women are more spread out around the mean. That is, there is more variability in the finishing times of women who ran the marathon compared to the finishing times of men who ran the marathon.

Answer:

(a) The standardized score for Clay is approximately $1.62$. It means Clay's finishing time is about 1.62 standard deviations above the mean finishing time for men.
(b) The proportion of women who had a finishing time less than Kathy's is approximately $0.6591$.
(c) The finishing times of women are more spread out around the mean compared to men, indicating more variability in women's finishing times.