You might need: Calculator, Z table
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 80 seconds and a 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. 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.
P (Yuki faster)

In this exercise we have to use knowledge about probability to calculate the probability of faster time, so we have to:

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

When the distribution is normal, we use the z-score formula. In a set with mean  and standard deviation , the zscore of a measure X is given by:

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 pvalue, we get the probability that the value of the measure is greater than X.

Find the probability that Yuki's time is faster than Zana's. That is, the subtraction of Yuki's time by Zana's time has to be higher than 0. When we subtract normal distributions, the mean is the subtraction of the means. So:

The standard deviation is the square root of the sum of the variances. So:

We have to find 1 subtracted by the p-value of Z when X = 0. So:

Has a p-value of 0.7611:

See more about probability at link

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

Step-by-step explanation:

When the distribution is normal, we use the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

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 pvalue, we get the probability that the value of the measure is greater than X.

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

That is, the subtraction of Yuki's time by Zana's time has to be higher than 0.

When we subtract normal distributions, the mean is the subtraction of the means. So

The standard deviation is the square root of the sum of the variances. So

We have to find 1 subtracted by the pvlaue of Z when X = 0. So

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.

c

step-by-step explanation:

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