Based on the empirical rule, we could predict that the percentage of gym members aged between 26 and 70 is about 95%
Further explanation
The probability of an event is defined as the possibility of an event occurring against sample space.
Let us tackle the problem.
[tex]\texttt{ }[/tex]
Empirical Rule of Normal Distribution :
[tex]\left[\begin{array}{ccc}Interval&Empirical Rule\\\overline{x} \pm\sigma&68 \%\\\overline{x} \pm 2\sigma&95 \%\\\overline{x} \pm 3\sigma&99.7 \%\end{array}\right][/tex]
where:
[tex]\overline{x}[/tex] = mean
[tex]\sigma[/tex] = standard deviation
[tex]\texttt{ }[/tex]
The ages of the members of a gym have a bell-shaped distribution with a mean of 48 years and a standard deviation of 11 years.
[tex]\overline{x} = 48[/tex] and [tex]\sigma = 11[/tex]
The percentage of gym members aged between 26 and 70 :
[tex]26 = 48 - 22 = 48 - 2(11) = \overline{x} - 2\sigma[/tex]
[tex]70 = 48 + 22 = 48 + 2(11) = \overline{x} + 2\sigma[/tex]
Based on Empirical Rule we can conclude that :
The percentage of gym members aged between 26 and 70 [tex]( \overline{x} \pm 2\sigma)[/tex] will be approximately 95%
[tex]\texttt{ }[/tex]
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Answer details
Grade: High School
Subject: Mathematics
Chapter: Probability
Keywords: Probability , Sample , Space , Six , Dice , Die , Normal , Distribution , Bell - Shaped , Empirical Rule , Prediction , Standard Deviation , Mean , Gym , Members , Ages , Rule , Average
