Question

Yuki and Zana are on a swimming team. They often compete against each other in the 100 meter freestyle race. Yuki's times in this race are normally distributed with a mean of 808080 seconds and a standard deviation of 4.24.24, point, 2 seconds. Zana's times are also normally distributed with a mean of 858585 seconds and a standard deviation of 5.65.65, point, 6 seconds. We can assume that their times are independent.

Suppose we choose a random 100 meter freestyle race and calculate the difference between their times.
Find the probability that Yuki's time is faster than Zana's.
You may round your answer to two decimal places.

Answer 1

0.2389 = 23.89% probability that Yuki's time is faster than Zana's.

Explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

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

`Z = (X - \mu)/(\sigma)`

The Z-score 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. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Subtraction of normal variables:

When normal variables are subtracted, the mean is the subtraction of the means, while the standard deviation is the square root of the sum of the variances.

Yuki's times in this race are normally distributed with a mean 80 seconds and standard deviation of 4.2 seconds. Zana's times are also normally distributed with a mean of 85 seconds and a standard deviation of 5.6 seconds.

Yuki's is faster than Zana if the subtraction of Yuki by Zana is larger than 0.

The mean is:

`\mu = 80 - 85 = -5`

The standard deviation is:

`\sigma = √(4.2^2 + 5.6^2) = 7`

Find the probability that Yuki's time is faster than Zana's.

This is P(X > 0), which is 1 subtracted by the pvalue of Z when X = 0.

`Z = (X - \mu)/(\sigma)`

`Z = (0 - (-5))/(7)`

`Z = 0.71`

`Z = 0.71` has a pvalue of 0.7611

1 - 0.7611 = 0.2389

0.2389 = 23.89% probability that Yuki's time is faster than Zana's.