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A 13-year-old girl who runs the mile in 5.5 minutes constantly teases her 9-year-old sister, who can only run the mile in 8 minutes. The 9-year-old sister, a budding statistics whiz, is out to prove her sister wrong. The average time it takes to run a mile for a 13-year-old girl is 9 minutes, with a standard deviation of 4 minutes. The average time it takes for a 9-year-old girl to run the mile is 11 minutes, with a standard deviation of 3. Assume running time for a mile is normally distributed. In a fair comparison, who is the faster runner? Justify your answer

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Answer to a math question A 13-year-old girl who runs the mile in 5.5 minutes constantly teases her 9-year-old sister, who can only run the mile in 8 minutes. The 9-year-old sister, a budding statistics whiz, is out to prove her sister wrong. The average time it takes to run a mile for a 13-year-old girl is 9 minutes, with a standard deviation of 4 minutes. The average time it takes for a 9-year-old girl to run the mile is 11 minutes, with a standard deviation of 3. Assume running time for a mile is normally distributed. In a fair comparison, who is the faster runner? Justify your answer

Expert avatar
Seamus
4.9
98 Answers
To determine who is the faster runner in a fair comparison, we will standardize the running times for both ages using z-scores and compare the resulting values.

Let's start with the 13-year-old girl:
Given:
Mean running time for a 13-year-old girl ($\mu_{13}$) = 9 minutes
Standard deviation for a 13-year-old girl ($\sigma_{13}$) = 4 minutes
Running time of the 13-year-old girl = 5.5 minutes

Calculating the z-score for the 13-year-old girl:
z = \frac{x - \mu}{\sigma}
z_{13} = \frac{5.5 - 9}{4}
z_{13} = \frac{-3.5}{4}
z_{13} = -0.875

Next, let's calculate for the 9-year-old sister:
Given:
Mean running time for a 9-year-old girl ($\mu_9$) = 11 minutes
Standard deviation for a 9-year-old girl ($\sigma_9$) = 3 minutes
Running time of the 9-year-old girl = 8 minutes

Calculating the z-score for the 9-year-old girl:
z = \frac{x - \mu}{\sigma}
z_9 = \frac{8 - 11}{3}
z_9 = \frac{-3}{3}
z_9 = -1

Comparing the z-scores, we see that the z-score for the 9-year-old girl is lower than the z-score for the 13-year-old girl. This means that the 9-year-old girl is the faster runner when compared to the average performance of her age group, according to the given data.

Therefore, in a fair comparison, the 9-year-old girl is the faster runner between the two sisters.

\boxed{9\text{-year-old girl is the faster runner.}}

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