USING THE EMPIRICAL RULE AND MAKING INFERENCES

A web site for prospective students at a university states that fulltime students spend an average of \$250 for textbooks each semester. The standard deviation of the amounts fulltime students spend for textbooks per semester is known to be \$50.

A. If the \$250 value given by the university is correct, do many students spend less than \$150 on textbooks each semester? Explain.

B. If the \$250 value given by the university is correct, what are the largest and smallest amounts virtually all students spend on textbooks per semester? Explain.

C. While buying your books at the bookstore, you notice that the student ahead of you spends \$480 for textbooks that semester. Does this cause you to suspect that the \$250 value reported by the university is incorrect? Explain.

SOLUTION

According to the university, the average amount a student spends on textbooks per semester is m = \$250. It has been stated that the standard deviation of all such expenditures is s = \$50. To answer any questions about the distribution of the amounts spent on textbooks when the values of the mean and standard deviation are known, the Empirical Rule can be used. To do this the intervals two and three standard deviations from the mean are calculated as follows:

m - 2s = \$250 – 2(\$50) = \$250 - \$100 = \$150

m + 2s = \$250 + 2(\$50) = \$250 + \$100 = \$350

m - 3s = \$250 – 3(\$50) = \$250 - \$150 = \$100

m + 3s = \$250 + 3(\$50) = \$250 + \$150 = \$400

Presuming the \$250 value given by the university is correct, the Empirical Rule tells us that most students will spend between \$150 and \$350 for textbooks each semester; all, or almost all, students will spend between \$100 and \$400 for textbooks. We may now use this information to answer the questions asked above.

A. Since most students will spend between \$150 and \$350 for textbooks each semester, few of the students would spend less than \$150.

B. Since virtually all students will spend between \$100 and \$400 for textbooks, these are the smallest and largest amounts we would expect to see students spend.

C. Since virtually all the students should spend between \$100 and \$400 if the value of \$250 reported by the university is correct, it is virtually impossible to observe a student spending \$480 for textbooks.  Thus, we will conclude that the mean amount spent on textbooks of \$250 reported by the university is incorrect. Observing a student spending \$480 supports the inference that students spend an average of more than \$250 per semester.

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