It is believed that 25% of U.S. homes have a direct satellite television receiver. How large a sample is necessary to estimate the true population of homes that have a direct satellite television receiver with 95% confidence and within 5 percentage points?

Answer:
Given:

The Population Proportion $ (p) = 0.25 $
The Margin Of Error $ (E) = 0.05 $
The Confidence Interval Level $ = 95% $

Solution:
The Significance Level $ (\alpha) : $
$ \alpha = 1 – 0.95 $
$ = 0.05 $

The Critical Value $ (z_c): $
$ z_c = z_{\alpha/2} $
$ = z_{0.05/2} $
$ = 1.96 $

→ Here we use a margin of error formula to find out the sample size

$ E = z_c \times \sqrt{\frac{p \times (1 – p)}{n}} $

$ 0.05 = 1.96 \times \sqrt{\frac{0.25 \times (1 – 0.25)}{n}} $

$ n = 288.12 $

$ n \approx 288 $

Final Answer:
The sample size $ (n) = 288 $