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Chapter 10: Problem 93

Maria and Ellen both specialize in throwing the javelin. Maria throws the javelin a mean distance of 200 feet with a standard deviation of 10 feet, whereas Ellen throws the javelin a mean distance of 210 feet with a standard deviation of 12 feet. Assume that the distances each of these athletes throws the javelin are normally distributed with these population means and standard deviations. If Maria and Ellen each throw the javelin once, what is the probability that Maria's throw is longer than Ellen's?

Short Answer

Expert verified
To calculate the exact probability, use standard normal distribution tables with the calculated Z-score.

Step by step solution

01

Identify Given Information

Maria's javelin throw is normally distributed with a mean distance of 200 feet and standard deviation of 10 feet. Ellen's is also normally distributed with a mean distance of 210 feet and standard deviation of 12 feet.
02

Formulating the Problem

The aim is to find out the probability that Maria's throw is longer than Ellen's. This can be formulated as finding the probability that the difference \( (X - Y) \) is greater than 0, where X is Maria's throw and Y is Ellen's.
03

Understanding the Normal Distribution

The difference between Maria's and Ellen's throws is also normally distributed. The mean difference is \( E(X-Y) = E(X) - E(Y) = 200 - 210 = -10 \). The variance of the difference is \( Var(X-Y) = Var(X) + Var(Y) = 10^2 + 12^2 = 244 \). Hence, the standard deviation is \( \sqrt{244} \)
04

Computing the Z-score

To know whether Maria's throw will be longer than Ellen's, we need to find \( P(X - Y > 0) \). However, normal distribution tables are typically standardized to Z-scores, so we need to transform the difference into a standard normal Z-value using the formula \( Z = \frac{X - mean}{standard deviation} \), in this case, \( Z = \frac{0 - (-10)}{\sqrt{244}} \)
05

Finding the Probability

The Z-score gives us the ability to use the standard normal distribution table to find the corresponding probability, which is the solution.

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Most popular questions from this chapter

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