Use the Empirical Rule to Describe Data That Are Bell Shaped
-Health care issues are receiving much attention in both academic and political arenas. A sociologist recently conducted a survey of citizens over 60 years of age whose net worth is too high to qualify for government health care but who have no private health insurance. The ages of 25 uninsured senior citizens were as follows:
$\begin{array} { l l l l l l l l l l l l l } 68 & 73 & 66 & 76 & 86 & 74 & 61 & 89 & 65 & 90 & 69 & 92 & 76 \end{array}$
$\begin{array} { l l l l l l l l l l l l } 62 & 81 & 63 & 68 & 81 & 70 & 73 & 60 & 87 & 75 & 64 & 82 \end{array}$
Suppose the mean and standard deviation are $74.0$ and $9.7$, respectively. If we assume that the distribution of ages is bell shaped, what percentage of the respondents will be between $64.3$ and $93.4$ years old?
A) approximately $81.5 \%$
B) approximately $68 \%$
C) approximately $95 \%$
D) approximately $83.9 \%$

Question

Quizplus · Accepted Answer

The Empirical Rule states that for a bell-shaped distribution, approximately 68% of the data will fall within one standard deviation of the mean, 95% of the data will fall within two standard deviations of the mean, and 99.7% of the data will fall within three standard deviations of the mean. In this case, the mean is 74.0 and the standard deviation is 9.7. So, 68% of the data will fall between 64.3 and 83.7, 95% of the data will fall between 54.6 and 93.4, and 99.7% of the data will fall between 44.9 and 103.1. Therefore, approximately 95% of the respondents will be between 64.3 and 93.4 years old.

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