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 90% confidence and within 3 percentage points?

Answer:
Given:

The Population Proportion $ (p) = 0.25 $
The Margin Of Error $ (E) = 0.03 $
The Confidence Interval Level $ = 90% $

Solution:
The Significance Level $ (\alpha) : $
$ \alpha = 1- 0.90 $
$ = 0.10 $

The Critical Value $ (z_c): $
$ z_c = z_(\alpha/2) $
$ = z_(0.10/2) $
$ = 1.645 $

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

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

$ 0.03 = 1.645 \times \sqrt{\frac{0.25 \times (1 – 0.25)}{n}} $

$ n = 563.76 $

$ n \approx 564 $

Final Answer:
The sample size $ (n) = 564 $