Step-by-step explanation:
In statistics, the empirical rule states that for a normally distributed random variable,
In mathematical notation, as shown in the figure below (for a standard normal distribution), the empirical rule is described as
[tex]\Phi(\mu \ - \ \sigma \ \leq X \ \leq \mu \ + \ \sigma) \ = \ 0.6827 \qquad (4 \ \text{s.f.}) \\ \\ \\ \Phi(\mu \ - \ 2\sigma \ \leq X \ \leq \mu \ + \ 2\sigma) \ = \ 0.9545 \qquad (4 \ \text{s.f.}) \\ \\ \\ \Phi}(\mu \ - \ 3\sigma \ \leq X \ \leq \mu \ + \ 3\sigma) \ = \ 0.9973 \qquad (4 \ \text{s.f.})[/tex]
where the symbol [tex]\Phi[/tex] (the uppercase greek alphabet phi) is the cumulative density function of the normal distribution, [tex]\mu[/tex] is the mean and [tex]\sigma[/tex] is the standard deviation of the normal distribution defined as [tex]N(\mu, \ \sigma)[/tex].
According to the empirical rule stated above, the interval that contains the prices of 99.7% of college textbooks for a normal distribution [tex]N(113, \ 12)[/tex],
[tex]\Phi(113 \ - \ 3 \ \times \ 12 \ \leq \ X \ \leq \ 113 \ + \ 3 \ \times \ 12) \ = \ 0.9973 \\ \\ \\ \-\hspace{1.75cm} \Phi(113 \ - \ 36 \ \leq \ X \ \leq \ 113 \ + \ 36) \ = \ 0.9973 \\ \\ \\ \-\hspace{3.95cm} \Phi(77 \ \leq \ X \ \leq \ 149) \ = \ 0.9973[/tex]
Therefore, the price of 99.7% of college textbooks falls inclusively between $77 and $149.
