A nationwide test taken by high school sophomores and juniors has three sections

Question

A nationwide test taken by high school sophomores and juniors has three sections, each scored on a scale of to . In a recent year, the national mean score for the writing section was . 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 within 3 standard deviations of the mean, 47.8. (b) Suppose that the distribution is bell-shaped. If approximately 95% of the scores lie between 29.0 and 66.6 , then the approximate value of the standard deviation for the distribution, according to the empirical rule, is _____.

Explanation / Answer

a.
Chebyshev's theorem states:
At least 1 - 1/ k^2 * 100 % of any set of data will lie within k standard
deviations of the mean (where k>1)
Your problem has k = 3, so we substitute in Chebyschev's theorem:
At least (1 - 1/3^2) * 100 % of any set of data will lie within 3 standard
deviations of the mean (where 3 > 1)
At least (1 - 1/9) * 100% of of any set of data will lie within 3 standard
deviations of the mean (where 3 > 1)
At least of any set of data will lie within 3 standard
deviations of the mean (where 3 > 1)
At least 88.88% of any set of data will lie within 3 standard
deviations of the mean (where 3 > 1)
At least ~85% of any set of data will lie within 2.5 standard deviations
of the mean ((where 3 > 1)
b.
About 95% of the area under the normal curve is within two standard deviation of the mean. i.e. (u ± 2s.d)
So to the given normal distribution about 95% of the observations lie in between [ 29.0 , 66.6 ]                  

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