A researcher would like to estimate p, the proportion of U.S. adults who support recognizing civil unions between gay or lesbian couples.If the researcher would like to be 95% sure that the obtained sample proportion would be within 1.5% of p (the proportion in the entire population of U.S. adults), what sample size should be used? 17,778 4,445 1,112 67 45
Question
A researcher would like to estimate p, the proportion of U.S. adults who support recognizing civil unions between gay or lesbian couples.If the researcher would like to be 95% sure that the obtained sample proportion would be within 1.5% of p (the proportion in the entire population of U.S. adults), what sample size should be used? 17,778 4,445 1,112 67 45
Solution
The sample size needed for a proportion can be calculated using the formula:
n = (Z^2 * p * (1-p)) / E^2
Where:
- Z is the z-value from the standard normal distribution corresponding to the desired confidence level (1.96 for 95% confidence level)
- p is the estimated proportion of the population (0.5 if unknown)
- E is the desired margin of error (0.015 in this case)
If we don't know the proportion (p), we use 0.5 because it gives us the maximum possible sample size, ensuring a sufficient sample.
Substituting the values into the formula:
n = (1.96^2 * 0.5 * (1-0.5)) / 0.015^2 n = 3.8416 * 0.25 / 0.000225 n = 0.9604 / 0.000225 n = 4266.222
So, the required sample size is approximately 4267 (always round up to ensure the margin of error is not exceeded).
Therefore, the closest answer to the options given is 4,445.
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