Questions: In preparing a report on the economy, we need to estimate the percentage of businesses that plan to hire additional employees in the next 60 days. a) How many randomly selected employers must we contact in order to create an estimate in which we are 95% confident with a margin of error of 5%? b) Suppose we want to reduce the margin of error to 3%. What sample size will suffice? c) Why might it not be worth the effort to try to get an interval with a margin of error of 1%? a) A sample size of is needed. (Round up to the nearest whole number.)

In preparing a report on the economy, we need to estimate the percentage of businesses that plan to hire additional employees in the next 60 days.
a) How many randomly selected employers must we contact in order to create an estimate in which we are 95% confident with a margin of error of 5%?
b) Suppose we want to reduce the margin of error to 3%. What sample size will suffice?
c) Why might it not be worth the effort to try to get an interval with a margin of error of 1%?
a) A sample size of is needed.
(Round up to the nearest whole number.)

Transcript text: In preparing a report on the economy, we need to estimate the percentage of businesses that plan to hire additional employees in the next 60 days. a) How many randomly selected employers must we contact in order to create an estimate in which we are 95\% confident with a margin of error of $5 \%$ ? b) Suppose we want to reduce the margin of error to $3 \%$. What sample size will suffice? c) Why might it not be worth the effort to try to get an interval with a margin of error of $1 \%$ ? a) A sample size of $\square$ is needed. (Round up to the nearest whole number.)

Solution

Solution Steps

Step 1: Determine Sample Size for 5% Margin of Error

To estimate the percentage of businesses that plan to hire additional employees with a margin of error of $5\%$ at a $95\%$ confidence level, we calculated the required sample size. The result is:

\[ n_{5\%} = 384 \]

Step 2: Determine Sample Size for 3% Margin of Error

To reduce the margin of error to $3\%$, we recalculated the sample size. The result is:

\[ n_{3\%} = 1067 \]

Step 3: Determine Sample Size for 1% Margin of Error

Finally, we calculated the sample size needed for a margin of error of $1\%$. The result is:

\[ n_{1\%} = 9604 \]