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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 50.8, with a standard deviation of 9.6

 According to Chebyshev's theorem, at least ?56%75%84%89% of the scores lie between 26.8 and 74.8 .

 

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(b) According to Chebyshev's theorem, at least ?56%75%84%89% of the scores lie between 31.6 and 70.0 .
 

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(c) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately ?68%75%95%99.7%of the scores lie between 31.6 and 70.0 .
 

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(d) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 99.7% of the scores lie between 

 

 

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Answer 1

(a) The z score for 26.8 to 74.8 is (26.8-50.8)/9.6=-24/9.6=-2.5 and (74.8-50.8)/9.6=24/9.6=2.5. So z^2=6.25. The z score measures how many standard deviations there are from the mean, so the range given is 2.5 SD's from the mean. Chebyshev's says (1-1/z^2) is the minimum probability that the scores are between the limits. Put z=2.5 and we get 0.84 or 84%.

(b) The z score for 31.6 to 70.0 is (31.6-50.8)/9.6=-2 and (70.0-50.8)/9.6=2, so z^2=4. According to Chebyshev, the minimum probability is 0.75 or 75%.

(c) The empirical rule or 1-2-3 rule for rough estimates of the probability gives 68-95-99.7 for 1, 2, 3 SD's from the mean so, 95% of scores are expected to lie between 31.6 and 70.0.

(d) The normal or bell distribution gives 0.9772 or 97.72% for z=2. For 99.7%, z=2.75. This is equivalent to 2.75*9.6=26.4, corresponding to 24.4 and 77.2, where 99.7%of scores are expected to lie.