Question

Northeast Commutes. Refer to Exercise $8.129.$

a. Determine and interpret a $90\%$ upper confidence bound for the mean commute time of all commuters in Washington, DC.

b. Compare your one-sided confidence interval in part (a) to the (twosided) confidence interval found in Exercise $8.129.$

Short answer

The mean commute time of all commuters in Washington, DC is lesser than $30.37$

Step-by-step solution

Given information

The following commute times, in minutes, were obtained from a sample of $30$ commuters in the Washington, D.C., area.

Concept

Upper confidence bound$=\overline{x}+t_{\alpha }\left(\frac{s}{\sqrt{n}}\right)$

Calculation

Determine $90\%$upper confidence bound for all commuters in Washington, DC's mean commute time.

Consider $\overline{x}=27.97,n=30$, and $s=10.04$

The required value of $t_{\frac{a}{2}}$for $90\%$confidence $29=(30-1)$comes from "Table IV Values of $t_{\alpha }$" $1.311$is the number of degrees of freedom.

The upper confidence bound formula is as follows:

Upper confidence bound $=\overline{x}+t_{\alpha }\left(\frac{s}{\sqrt{n}}\right)$

Therefore,

$$\begin{gathered} \overline{x}+t_{\frac{\alpha }{2}}\left(\frac{s}{\sqrt{n}}\right)=27.97+1.311\left(\frac{10.04}{\sqrt{30}}\right) \\ =27.97+1.311(1.833) \\ =27.97+2.403 \\ =30.37 \end{gathered}$$

As a result, the average commuting time in Washington, DC is less than $30.37$