Question

a survey of 70 students found that 29% were in favor of raising tuition to build a new recreation center. the standard deviation of the sample proportion is 9.5%. how large a sample (to the nearest person) would be required to reduce this standard deviation to 6.5%?

Answer 1

To reduce the standard deviation from 9.5% to 6.5%, a sample size of approximately 315 students would be required.

Step-by-step explanation:

To determine the sample size needed to reduce the standard deviation from 9.5% to 6.5%, we can use the formula for sample size calculation for proportions. The formula is:

n = (z * sqrt(p * (1 - p)) / E)^2

where n is the sample size, z is the z-score corresponding to the desired level of confidence, p is the estimated proportion, and E is the desired margin of error.

In this case, the standard deviation is expressed as a percentage, so we need to divide it by 100 to convert it to a proportion. Thus, the estimated proportion is 0.095 and the desired margin of error is 0.065. Using a z-score of 1.96 (corresponding to a 95% level of confidence), we can substitute these values into the formula to calculate the sample size:

n = (1.96 * sqrt(0.29 * (1 - 0.29)) / 0.065)^2

By solving this equation, we find that the sample size needed is approximately 315 students.