Chapter 11, Problem 63E

Solution

To calculate: The probability that a tune-up will take more than 2 hours and the probability that a tune-up will take less than 66 minutes.

The probability that a tune-up will take more than 2 hours or 120 minutes is 0.16 and the probability that a tune-up will take less than 66 minutes is 0.025.

Given:The distribution of time taken for a tune-up follows a normal distribution with mean $\mu =102$ and standard deviation $\sigma =18$ .

Concept used:

The Empirical rule states that within one standard deviation of the mean, there is about 68% of the data, within two standard deviations of the mean, there is about 95% of the data, and within three standard deviations of the mean, there is about 99.7% of the data.

  

Calculation:

With reference to the normal distribution curve and empirical rule as shown above the one standard deviation and two standard deviations away from points are calculated as shown below

  $\begin{array}{l}\mu -\sigma =102-18=84\\ \mu +\sigma =102+18=120\end{array}$$\begin{array}{l}\mu -2\sigma =102-2\times 18=66\\ \mu +2\sigma =102+2\times 18=138\end{array}$

Thus according to the empirical rule, 120 minutes is one standard deviation more than the mean, and the probability greater than the mean 102 is about 50% and the probability between the mean and 120 minutes is about 34%.

Thus the probability that it takes more than 120 minutes for a tune-up is calculated as shown below

  $\begin{array}{l}P(X>120)=P(X>102)-P(102<X<120)\\ P(X>120)=0.5-0.34\\ P(X>120)=0.16\end{array}$ .

The probability that it takes more than 120 minutes for a tune-up is 0.16.

Thus according to the empirical rule, 66 minutes is two standard deviations less than the mean, and the probability less than the mean 102 is about 50% and the probability between the mean and 66 minutes is about $34\%+13.5\%=47.5\%$ .

Thus the probability that it takes less than 66 minutes for a tune-up is calculated as shown below

  $\begin{array}{l}P(X<66)=P(X<102)-P(66<X<102)\\ P(X<66)=0.5-0.475\\ P(X<66)=0.025\end{array}$

The probability that it takes less than 66 minutes for a tune-up is 0.025.

Conclusion:With reference to the empirical rule, the probability that a tune-up takes more than 120 minutes is 0.16 and the probability that a tune-up takes less than 66 minutes is 0.025.