To calculate the confidence interval for the proportion of people who subscribe to cable TV, follow these steps:
1. Determine the sample proportion ([tex]\(\hat{p}\)[/tex]):
[tex]\[
\hat{p} = \frac{\text{subscribers}}{\text{sample size}} = \frac{255}{420} = 0.607
\][/tex]
2. Determine the standard error (SE) for the sample proportion:
[tex]\[
\text{SE} = \sqrt{\frac{\hat{p} (1 - \hat{p})}{\text{sample size}}} = \sqrt{\frac{0.607 \times (1 - 0.607)}{420}} \approx 0.024
\][/tex]
3. Identify the Z-score for the desired confidence level:
From the provided table, the Z-score for a 90% confidence level is:
[tex]\[
z_{0.05} = 1.645
\][/tex]
4. Calculate the margin of error (ME):
[tex]\[
\text{ME} = z \times \text{SE} = 1.645 \times 0.024 \approx 0.039
\][/tex]
5. Calculate the confidence interval:
[tex]\[
\text{Lower bound} = \hat{p} - \text{ME} = 0.607 - 0.039 = 0.568
\][/tex]
[tex]\[
\text{Upper bound} = \hat{p} + \text{ME} = 0.607 + 0.039 = 0.646
\][/tex]
6. Round the final answers to three decimal places:
[tex]\[
\text{Lower bound: } 0.568
\][/tex]
[tex]\[
\text{Upper bound: } 0.646
\][/tex]
So, the 90% confidence interval for the proportion of people who subscribe to cable TV is [tex]\([0.568, 0.646]\)[/tex].
