Final answer:
To determine the required sample size for a 95% confidence interval with a margin of error of at most 0.07 and a past support rate of 93%, the formula n = (Z^2 * p * (1-p)) / E^2 is used. The calculation provides a sample size slightly above 101, which means none of the provided options match exactly, suggesting a possible need to verify the options.
Step-by-step explanation:
Assuming that you wish to estimate a population proportion, p, with a certain margin of error and confidence level, the sample size required can be determined with the use of a formula derived from the principles of statistics. To calculate the sample size for a population proportion where the desired margin of error is at most 0.07 for a 95% confidence interval, given that past similar environmental initiatives have received 93% support, we apply the formula:
n = (Z^2 * p * (1-p)) / E^2
Where n is the sample size, Z is the Z-score associated with the desired confidence level, p is the estimated proportion of support, and E is the margin of error.
For a 95% confidence interval, the Z-score is typically 1.96. Thus, using p = 0.93 and E = 0.07, the calculation becomes:
n = (1.96^2 * 0.93 * (1 - 0.93)) / 0.07^2
This results in n approximately equal to 101. However, since sample size needs to be a whole number, we would need to round up to the next whole number to ensure the margin of error is met, which would be 102. Therefore, based on our calculations, none of the given options (A) 156, (B) 104, (C) 46, (D) 52 match our calculated value exactly. Please verify the options given.