Question

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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)

Answer 1

To find the probability that Yuki's time is faster than Zana's, calculate the z-score for Yuki's time and use the z-table to find the corresponding probability.

Step-by-step explanation:

To find the probability that Yuki's time is faster than Zana's, we need to compare their times using the normal distribution. We can calculate the z-score for Yuki's time and then use the z-table to find the probability.

1. First, calculate the z-score for Yuki's time:
2. Z = (Yuki's time - Yuki's mean) / Yuki's standard deviation
3. 
   
4. Z = (Yuki's time - 80) / 4.2
5. Next, use the z-score to find the probability using the z-table. Look up the z-score and find the corresponding area under the normal curve to the left of the z-score.
6. The probability that Yuki's time is faster than Zana's is the area under the normal curve to the left of Yuki's z-score.