Suppose the time to complete a 200-meter backstroke swim for female competitive swimmers is normally distributed with a mean μ = 141 seconds and a standard deviation σ = 7 seconds.Suppose the fastest 6% of female swimmers in the nation are offered college scholarships. In order to be given a scholarship, a swimmer must complete the 200-meter backstroke in no more than how many seconds? Give your answer in whole numbers.
Question
Suppose the time to complete a 200-meter backstroke swim for female competitive swimmers is normally distributed with a mean μ = 141 seconds and a standard deviation σ = 7 seconds.Suppose the fastest 6% of female swimmers in the nation are offered college scholarships. In order to be given a scholarship, a swimmer must complete the 200-meter backstroke in no more than how many seconds? Give your answer in whole numbers.
Solution
To solve this problem, we need to find the z-score that corresponds to the fastest 6% of swimmers.
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The z-score is a measure of how many standard deviations an element is from the mean. In a normal distribution, the fastest 6% of swimmers corresponds to the 94th percentile (since we start counting from the slowest times).
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We can use a z-table or a calculator to find the z-score that corresponds to the 94th percentile. The z-score is approximately 1.55.
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Once we have the z-score, we can use the formula for the z-score to find the time. The formula is:
Z = (X - μ) / σ
Where: Z is the z-score, X is the value we're looking for, μ is the mean, and σ is the standard deviation.
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We can rearrange the formula to solve for X:
X = Z * σ + μ
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Substituting the values we know:
X = 1.55 * 7 + 141
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Calculating the above expression gives us X ≈ 150.85 seconds.
So, a swimmer must complete the 200-meter backstroke in no more than 151 seconds (rounded to the nearest whole number) to be in the top 6% and be offered a college scholarship.
Similar Questions
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