To determine the sample size needed to estimate the proportion of U.S. households that subscribe to cable TV within 5 percentage points with a 95% level of confidence, we can use the formula for the sample size of a proportion:

$$ n = \left( \frac{Z^2 \cdot p \cdot (1 - p)}{E^2} \right) $$

Where:
- $ n $ is the sample size.
- $ Z $ is the z-value corresponding to the desired confidence level.
- $ p $ is the estimated proportion (0.60 in this case).
- $ E $ is the margin of error (0.05 in this case).

Step-by-step solution:

1. **Determine the z-value for a 95% confidence level:**
The z-value for a 95% confidence level is 1.96 (from the z Distribution Table).

2. **Substitute the values into the formula:**
$$ n = \left( \frac{1.96^2 \cdot 0.60 \cdot (1 - 0.60)}{0.05^2} \right) $$

3. **Calculate the numerator:**
$$ 1.96^2 = 3.8416 $$
$$ 0.60 \cdot (1 - 0.60) = 0.60 \cdot 0.40 = 0.24 $$
$$ 3.8416 \cdot 0.24 = 0.922 $$

4. **Calculate the denominator:**
$$ 0.05^2 = 0.0025 $$

5. **Divide the numerator by the denominator:**
$$ n = \frac{0.922}{0.0025} = 368.8 $$

6. **Round to the next whole number:**
$$ n \approx 369 $$

Therefore, you should sample 369 households to estimate the proportion of U.S. households that subscribe to cable TV within 5 percentage points with a 95% level of confidence.

Question

To determine the sample size needed to estimate the proportion of U.S. households that subscribe to cable TV within 5 percentage points with a 95% level of confidence, we can use the formula for the sample size of a proportion:

$$ n = \left( \frac{Z^2 \cdot p \cdot (1 - p)}{E^2} \right) $$

Where:
- $ n $ is the sample size.
- $ Z $ is the z-value corresponding to the desired confidence level.
- $ p $ is the estimated proportion (0.60 in this case).
- $ E $ is the margin of error (0.05 in this case).

Step-by-step solution:

1. **Determine the z-value for a 95% confidence level:**
   The z-value for a 95% confidence level is 1.96 (from the z Distribution Table).

2. **Substitute the values into the formula:**
   $$ n = \left( \frac{1.96^2 \cdot 0.60 \cdot (1 - 0.60)}{0.05^2} \right) $$

3. **Calculate the numerator:**
   $$ 1.96^2 = 3.8416 $$
   $$ 0.60 \cdot (1 - 0.60) = 0.60 \cdot 0.40 = 0.24 $$
   $$ 3.8416 \cdot 0.24 = 0.922 $$

4. **Calculate the denominator:**
   $$ 0.05^2 = 0.0025 $$

5. **Divide the numerator by the denominator:**
   $$ n = \frac{0.922}{0.0025} = 368.8 $$

6. **Round to the next whole number:**
   $$ n \approx 369 $$

Therefore, you should sample 369 households to estimate the proportion of U.S. households that subscribe to cable TV within 5 percentage points with a 95% level of confidence.

Knowee AI · Accepted Answer