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 50.4, with a standard deviation of 9.1. 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____of the scores lie between 23.1 and 77.7.

B. According to chebyshev's theorem, at least_____of the scores lie between 32.2 and 68.6.

C. Suppose that the distribution is bell-shaped. According to the empirical rule, approximately_____of the scores lie between 32.2 and 68.6.

D. Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 68% of the scores lie between____and_____.

Answer 1

According to chebyshev's inequality theorem, at least 1 - 1/k² of the observations must lie within k standard deviations from the mean.

a) 23.1 = 50.4 - 3*9.1 that is 3 standard deviations less than mean.
77.7 = 50.4 + 3*9.1 that is 3 standard deviations less than the mean.

1 - 1/3² = 8/9 = 88.89%

Therefore 88.89% is the required % here.

b) 32.2 = 50.4 - 2*9.1 that is 2 standard deviations less than mean.
68.6 = 50.4 + 2*9.1 that is 2 standard deviations less than the mean.

1 - 1/2² = 3/4 = 75%

Therefore 75% is the required % here.

c) For a normal distribution 95% of the observations must lie within 2 standard deviations of the mean. Therefore 95% is the required % here.

d) For a normal distribution 68% of the observations must lie within 1 standard deviation of the mean. Therefore 68% of the observations lies between:

50.4 - 9.1 = 41.3 and
50.4 + 9.1 = 59.5