Question

Math

Posted 6 months ago

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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.
$P(\text { Yuki faster }) \approx$
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Answer from Sia

Posted 6 months ago

Solution by Steps

step 1

Define the random variables for Yuki's and Zana's race times. Let $Y$ represent Yuki's time and $Z$ represent Zana's time

step 2

Given that $Y \sim N(80, 4.2^2)$ and $Z \sim N(85, 5.6^2)$, we want to find the probability $P(Y < Z)$

step 3

Define a new random variable $D = Y - Z$ which represents the difference in race times. Since $Y$ and $Z$ are independent, the mean of $D$ is $\mu_D = \mu_Y - \mu_Z$ and the variance of $D$ is $\sigma_D^2 = \sigma_Y^2 + \sigma_Z^2$

step 4

Calculate the mean and variance of $D$: $\mu_D = 80 - 85 = -5$ and $\sigma_D^2 = 4.2^2 + 5.6^2$

step 5

Calculate the standard deviation of $D$: $\sigma_D = \sqrt{4.2^2 + 5.6^2}$

step 6

Convert the problem to a standard normal distribution problem by finding the z-score for $D = 0$: $z = \frac{0 - (-5)}{\sigma_D}$

step 7

Use the standard normal distribution table or a calculator to find the probability $P(D > 0)$, which is the same as $P(Y < Z)$

step 8

Report the probability rounded to two decimal places

Answer

The probability that Yuki's time is faster than Zana's is approximately [insert probability here rounded to two decimal places]

Key Concept

Difference of Independent Normal Random Variables

Explanation

When finding the probability that one normally distributed random variable is less than another, we can define a new random variable as the difference of the two. If the original variables are independent, the new variable is also normally distributed with its mean and variance being the difference of the means and the sum of the variances, respectively. We then standardize this new variable to find the probability using the standard normal distribution.

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