Question

Among the thirty largest U.S. cities, the mean one-way commute time to work is 25.8 minutes. The longest one-way travel time is in New York City, where the mean time is 37.7 minutes. Assume the distribution of travel times in New York City follows the normal probability distribution and the standard deviation is 6.7 minutes. Use Appendix B.3.

1. What percent of the New York City commutes are for less than 27 minutes? (Round your intermediate calculations and final answer to 2 decimal places.)

2. What percent are between 27 and 33 minutes? (Round your intermediate calculations and final answer to 2 decimal places.)

3. What percent are between 27 and 47 minutes? (Round your intermediate calculations and final answer to 2 decimal places.)

Answer 1

$\boldsymbol{SOLUTION}$

$\mathbf{given:}\;mean\;(\mu)=\mathbf{37.7},\;standard\;deviation\;(\sigma)=\mathbf{6.7}$

$\mathbf{Let \;"X" \;be \;commuting \;time}$

$\mathbf{formula:}\;Z=\frac{X-\mu}{\sigma}$
${\color{Blue} \mathbf{a.}}\;P(X<27)\Rightarrow P\left ( \frac{X-\mu}{\sigma}<\frac{27-37.7}{6.7} \right )$

$=P(Z<-1.60)$

$=\mathbf{{\color{Red} 0.0548}}\;(\mathbf{from\;z-table})$

${\color{Blue} \mathbf{b.}}\;P(27<X<33)\Rightarrow P\left (\frac{27-37.7}{6.7}< \frac{X-\mu}{\sigma}<\frac{33-37.7}{6.7} \right )$

$\Rightarrow P(-1.60<Z<-0.70)$

$\Rightarrow P(Z<-0.70)-P(Z<-1.60)$

$\Rightarrow 0.2420-0.0548\;(\mathbf{from\;z-table})$

$\Rightarrow \mathbf{{\color{Red} 0.1872}}$

${\color{Blue} \mathbf{c.}}\;P(27<X<47)\Rightarrow P\left (\frac{27-37.7}{6.7}< \frac{X-\mu}{\sigma}<\frac{47-37.7}{6.7} \right )$

$\Rightarrow P(-1.60<Z<1.39)$

$\Rightarrow P(Z<1.39)-P(Z<-1.60)$

$\Rightarrow 0.9177-0.0548\;(\mathbf{from\;z-table})$

$\Rightarrow \mathbf{{\color{Red} 0.8629}}$