Chapter 19, Problem 30E

(a)

To determine

To explain how many randomly selected employers must we contact in order to create an estimate in which we are $98\%$ confident with margin of error of $5\%$ .

Answer to Problem 30E

  $542$ .

Explanation of Solution

It is given in the question that in preparing a report on the economy, we need to estimate the percentage of business that plan to hire additional employees in the next $60$ days. Thus, to find how many randomly selected employers must we contact in order to create an estimate in which we are $98\%$ confident with margin of error of $5\%$ , we have,

  $\begin{array}{l}0.05=2.326\times \sqrt{\frac{0.5\times 0.5}{n}}\\ \text{Solve for }n\\ \Rightarrow 0.0215=\sqrt{\frac{0.5\times 0.5}{n}}\\ \Rightarrow n=542\\ (\text{Assume that }p=0.5)\end{array}$

Thus, we need $542$ selected employers that we must contact in order to create an estimate in which we are $98\%$ confident with margin of error of $5\%$ .

(b)

To determine

To find out what sample size will suffice.

Answer to Problem 30E

  $1504$ .

Explanation of Solution

It is given in the question that in preparing a report on the economy, we need to estimate the percentage of business that plan to hire additional employees in the next $60$ days. Thus, to find sample size, we have,

  $\begin{array}{l}0.03=1.645\times \sqrt{\frac{0.5\times 0.5}{n}}\\ \text{Solve for }n\\ \Rightarrow 0.0129=\sqrt{\frac{0.5\times 0.5}{n}}\\ \Rightarrow n=1504\\ (\text{Assume that }p=0.5)\end{array}$

(c)

To determine

To explain what might it not be worth the effort to try to get an interval with a margin of error of only $1\%$ .

Explanation of Solution

It is given in the question that in preparing a report on the economy, we need to estimate the percentage of business that plan to hire additional employees in the next $60$ days. Thus, it might not be worth the effort to try to get an interval with a margin of error of only $1\%$ because we would need to sample a ridiculously large amount of people and this would take a lot of time and money.