Question

A nationwide test taken by high school sophomores and juniors has three sections, each scored on a scale of 20 to 80 . In a recent year, the national mean score for the writing section was 49.9 . Based on this information, complete the following statements about the distribution of the scores on the writing section for the recent year.

(a) According to Chebyshev's theorem, at least ?56%75%84%89% of the scores lie within 2 standard deviations of the mean,

49.9

.

(b) Suppose that the distribution is bell-shaped. If approximately 99.7% of the scores lie between 21.1 and 78.7 , then the approximate value of the standard deviation for the distribution, according to the empirical rule, is .

Answer 1

(a)

Chebyshev's inequality :

Let X be a random variable with finite expected value   and finite non-zero variance . Then for any real number k>0,

So, 75% of the scores lie within 2 standard deviations of the mean.

(b)

The empirical rule states that for a normal distribution nearly 99.7% fall within three standard deviations.