Question

The finishing times for a race happen to be normally distributed with a mean of 10.2 seconds and a standard deviation of 1.4 seconds. Find the cut off score for the top 10%. Interpret this result.

Answer 1

Using the normal distribution, it is found that the cut off score for the top 10% is of 8.4 seconds. It means that a person with a time of 8.4 seconds or lower is in the faster 10% of runners.

In a normal distribution with mean $\mu$ and standard deviation $\sigma$, the z-score of a measure X is given by:

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

• It measures how many standard deviations the measure is from the mean.  
• After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.

In this problem:

• Mean of 10.2 seconds, hence $\mu = 10.2$
• Standard deviation of 1.4 seconds, hence $\sigma = 1.4$.

The lower the times, the faster the runners are, hence, the cut off score for the top 10% is the 10th percentile, which is X when Z has a p-value of 0.1, so X when Z = -1.28.

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

$-1.28 = \frac{X - 10.2}{1.4}$

$X - 10.2 = -1.28(1.4)$

$X = 8.4$

The cut off score for the top 10% is of 8.4 seconds. It means that a person with a time of 8.4 seconds or lower is in the faster 10% of runners.

A similar problem is given at https://brainly.com/question/24663213