To determine the confidence interval for the proportion of cable subscribers who would be willing to pay extra for a new nature channel, follow these steps:
1. Identify the given data:
- Sample size ([tex]\(n\)[/tex]): [tex]\(85\)[/tex]
- Sample proportion ([tex]\(\hat{p}\)[/tex]): [tex]\(0.39\)[/tex] or [tex]\(39\%\)[/tex]
- [tex]\(z^*\)[/tex]-score for [tex]\(95\%\)[/tex] confidence level: [tex]\(1.96\)[/tex]
2. Calculate the margin of error ([tex]\(E\)[/tex]):
[tex]\[
E = z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\][/tex]
3. Compute the margin of error:
[tex]\[
E \approx 1.96 \times \sqrt{\frac{0.39 \times (1 - 0.39)}{85}}
\][/tex]
[tex]\[
E \approx 0.10369166056433002
\][/tex]
4. Determine the confidence interval ([tex]\(C\)[/tex]):
[tex]\[
C = \hat{p} \pm E
\][/tex]
5. Calculate the lower and upper bounds of the confidence interval:
[tex]\[
C_{\text{lower}} = \hat{p} - E = 0.39 - 0.10369166056433002 \approx 0.28630833943567
\][/tex]
[tex]\[
C_{\text{upper}} = \hat{p} + E = 0.39 + 0.10369166056433002 \approx 0.493691660564330
\][/tex]
6. Convert the bounds to percentages:
[tex]\[
C_{\text{lower percent}} \approx 28.63\%
\][/tex]
[tex]\[
C_{\text{upper percent}} \approx 49.37\%
\][/tex]
Therefore, to the nearest percent, the confidence interval for the proportion of cable subscribers who would be willing to pay extra for the new nature channel is approximately between [tex]\(29\%\)[/tex] and [tex]\(49\%\)[/tex].
