Suppose the scores on an exam are normally distributed with a mean μ = 75 points, and standard deviation σ = 8 points.Suppose that the top 4% of the exams will be given an A+. In order to be given an A+, an exam must earn at least what score? Report your answer in whole numbers.
Question
Suppose the scores on an exam are normally distributed with a mean μ = 75 points, and standard deviation σ = 8 points.Suppose that the top 4% of the exams will be given an A+. In order to be given an A+, an exam must earn at least what score? Report your answer in whole numbers.
Solution
To find the score that marks the top 4% of the exams, we need to find the z-score that corresponds to the top 4% in a standard normal distribution.
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First, we need to find the z-score that corresponds to the top 4% of the distribution. We can look this up in a standard normal distribution table, or use a calculator or software that can calculate it. The z-score that corresponds to the top 4% is approximately 1.75.
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Next, we use the z-score formula to find the corresponding score in the exam distribution. The formula is:
X = μ + Zσ
where X is the score we're looking for, μ is the mean, Z is the z-score, and σ is the standard deviation.
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Substituting the given values into the formula, we get:
X = 75 + 1.75*8
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Calculating the right side of the equation gives us X = 89.
So, in order to be given an A+, an exam must earn at least 89 points.
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