Chapter 23, Problem 16E

Parking II. Suppose that, for budget planning purposes, the city in Exercise 14 needs a better estimate of the mean daily income from parking fees.

a) Someone suggests that the city use its data to create a $95\%$ confidence interval instead of the $90\%$ interval first created. How would this interval be better for the city? (You need not actually create the new interval.)

b) How would the $95\%$ interval be worse for the planners?

c) How could they achieve an interval estimate that would better serve their planning needs?

d) How many days’ worth of data should they collect to have $95\%$ confidence of estimating the true mean to within $\$3$ ?

To determine

To explain how would this interval be better for the city.

Answer to Problem 16E

It will be narrower.

Explanation of Solution

It is given in the question that hoping to lure more shoppers downtown a city builds a new parking garage in the central business district. The city plans to pay for the structure through parking fees. During a two-month period, daily fees collected averaged $\$126$ , with a standard deviation of $\$15$ .

Thus, someone suggest that the city use its data to create a $95\%$ confidence interval instead of the $90\%$ interval first created. Thus, this interval would be better for the city because if the confidence level is increased, then we become more sure that the confidence interval contains the true population mean, which is better for the city.

To determine

To explain how the $95\%$ interval be worse for the planners.

Explanation of Solution

It is given in the question that hoping to lure more shoppers downtown a city builds a new parking garage in the central business district. The city plans to pay for the structure through parking fees. During a two-month period, daily fees collected averaged $\$126$ , with a standard deviation of $\$15$ . And all conditions are met.

Now, the $95\%$ interval be worse for the planners because the $95\%$ confidence interval will be wider and thus their actual budget for the build of the new public parking garage could vary more strongly. Thus, it will be worse for the case.

To determine

To explain how could they achieve an interval estimate that would better serve their planning needs.

Answer to Problem 16E

By increasing the sample size.

Explanation of Solution

It is given in the question that hoping to lure more shoppers downtown a city builds a new parking garage in the central business district. The city plans to pay for the structure through parking fees. During a two-month period, daily fees collected averaged $\$126$ , with a standard deviation of $\$15$ .

Thus, they could achieve an interval estimate that would better serve their planning needs because increase the sample size, because then the same level of confidence can be obtained while the confidence interval will become narrower.

To determine

To explain how many days’ worth of data should they collect to have $95\%$ confidence of estimating the true mean to within $\$3$ .

Answer to Problem 16E

The large sample size is $97$ .

Explanation of Solution

It is given in the question that hoping to lure more shoppers downtown a city builds a new parking garage in the central business district. The city plans to pay for the structure through parking fees. During a two-month period, daily fees collected averaged $\$126$ , with a standard deviation of $\$15$ .

Now, we have to find how many days’ worth of data should they collect to have $95\%$ confidence of estimating the true mean to within $\$3$ , this can be done as:

It is given,

  $\begin{array}{l}E=3\\ \sigma =15\end{array}$

Mow, the formula to calculate the sample size will be:

  $n={(\frac{z_{\alpha /2}\sigma }{E})}^2$

Thus, for confidence level, $1-\alpha =0.95$ , we have,

  $\begin{array}{l}{z}_{\alpha /2}=z_{0.025}\\ \Rightarrow z_{\alpha /2}=1.96\end{array}$

Now, the sample size is:

  $\begin{array}{l}n={(}^{\frac{z_{\alpha /2}\sigma }{E}}\\ ={(}^{\frac{1.96\times 15}{3}}\\ \approx 97\end{array}$

Thus, the large sample size is $97$ .