A score below four hundred

Question

Exam scores are normally distributed with a mean of 500 and a standard deviation of 100. Use the empirical (68-95-99.7) rule. What is the probability that a score falls below 400?

Accepted Answer

Recall the empirical rule in standard-deviation steps: - about 68% of values fall within 1 standard deviation of the mean, - about 95% within 2 standard deviations, - about 99.7% within 3 standard deviations. The value 400 is \frac{400 - 500}{100} = -1 standard deviation, so we want the low tail beyond 1 standard deviation below the mean. The rule says 68% of scores fall within 1 standard deviation (between 400 and 600). That leaves 1 - 0.68 = 0.32 outside the band, split evenly into the two tails by symmetry. The low tail (below 400) gets half: P(X < 400) = \frac{1 - 0.68}{2} = \frac{0.32}{2…

A score below four hundred

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