A public relations firm found only 9% of voters in a certain state are satisfied with their US senators. How large a sample of voters should be drawn so that the sample proportion of voters who are satisfied with their senators is approximately normally distributed?
To determine the sample size needed for the sample proportion of voters who are satisfied with their senators to be approximately normally distributed, we can use the formula for sample size in a proportion estimation:
n = (Z^2 * p * q) / E^2
Where:
- n is the required sample size
- Z is the Z-score corresponding to the desired level of confidence (e.g., 1.96 for a 95% confidence level)
- p is the estimated proportion of voters satisfied with their senators (9% or 0.09)
- q is the complement of p (q = 1 - p)
- E is the maximum margin of error (expressed as a proportion, typically 0.05)
Let's calculate the sample size using these variables:
Z = 1.96 (for a 95% confidence level)
p = 0.09 (9% or 0.09)
q = 1 - p = 1 - 0.09 = 0.91
E = 0.05
n = (1.96^2 * 0.09 * 0.91) / 0.05^2
Simplifying the equation:
n = (3.8416 * 0.0819) / 0.0025
n = 0.3139 / 0.0025
n ≈ 125.56
Therefore, a sample size of approximately 126 voters should be drawn in order to have the sample proportion of voters who are satisfied with their senators approximately normally distributed.