Question

Show all your work. Indicate clearly the methods you use, because you will be scored on the correctness of your methods as well 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. The finishing time for Clay was 289 minutes. Calculate and interpret the standardized score for Clayâs marathon time. Show your work.

Answer 1

Answer:

Clay's finishing time is 1.68 standard deviation above the mean finishing time for men.

Step-by-step explanation:

In statistics, a standardized score is the number of standard deviations an observation or data point is from the mean.

Let us consider a random variable, X that follows a normal distribution, N (µ, σ²).

Then Z is a standard normal variate with mean, E (Z) = 0 and Var (Z) = 1. That is,

$Z=\frac{x-\mu}{\sigma}\sim N(0, 1)$

This, z-score is known as the standardized score.

It is provided that the distribution of finishing time for men (say, X) was approximately normal with mean µ = 242 minutes and standard deviation σ = 29 minutes.

The finishing time for Clay was x = 289 minutes.

Compute Clay's z-score as follows:

$Z=\frac{x-\mu}{\sigma}$

   $=\frac{289-242}{29}\\\\=1.67857143\\\\\approx 1.68$

Thus, Clay's standardized score is 1.68.

That is, Clay's finishing time is 1.68 standard deviation above the mean finishing time for men.