To determine how many customers the company should survey to be 98% confident that the estimated proportion is within 5 percentage points of the true population proportion, we will use the formula for calculating sample size for proportion:
[tex]\[ n = \frac{z^2 \cdot p \cdot (1 - p)}{E^2} \][/tex]
where:
- [tex]\( z \)[/tex] is the z-value corresponding to the desired confidence level,
- [tex]\( p \)[/tex] is the estimated population proportion,
- [tex]\( E \)[/tex] is the margin of error.
Given:
- Confidence level ([tex]\( 1 - \alpha \)[/tex]) = 98% (corresponding z-value [tex]\( z_{0.01} = 2.326 \)[/tex] from the table),
- Margin of error ([tex]\( E \)[/tex]) = 0.05 (5 points in decimal form),
- Estimated population proportion ([tex]\( p \)[/tex]) = 0.5.
Using these values, we can set up our formula:
[tex]\[ n = \frac{(2.326)^2 \cdot 0.5 \cdot (1 - 0.5)}{(0.05)^2} \][/tex]
First, calculate [tex]\( z^2 \)[/tex]:
[tex]\[ z^2 = (2.326)^2 = 5.410276 \][/tex]
Next, calculate [tex]\( p \times (1 - p) \)[/tex]:
[tex]\[ p \times (1 - p) = 0.5 \times 0.5 = 0.25 \][/tex]
Then, compute the numerator:
[tex]\[ \text{Numerator} = 5.410276 \times 0.25 = 1.352569 \][/tex]
Now, calculate the denominator:
[tex]\[ \text{Denominator} = (0.05)^2 = 0.0025 \][/tex]
Finally, divide the numerator by the denominator to find [tex]\( n \)[/tex]:
[tex]\[ n = \frac{1.352569}{0.0025} = 541.0276 \][/tex]
Since the sample size must be a whole number, we round up:
[tex]\[ n \approx 542 \][/tex]
Therefore, the company should survey 542 customers to be 98% confident that the estimated proportion is within 5 percentage points of the true population proportion.
The correct answer is:
[tex]\[ 542 \text{ customers} \][/tex]
