A random sample of 100 voters is taken to estimate the proportion of a state's electorate in favor of increasing the gasoline tax to provide additional revenue for highway repairs. Suppose that it is decided that a sample of 100 voters is too small to provide a sufficiently reliable estimate of the population proportion. It is required instead that the probability that the sample proportion differs from the population proportion (whatever its value) by more than 0.03 should not exceed 0.075. How large a sample is needed to guarantee that this requirement is met?
To calculate the minimum sample size needed to guarantee that the requirement is met, we can use the formula for calculating the sample size needed for estimating population proportion with a specified margin of error:
n = (Z^2 * p * (1-p))/(E^2)
where:
n = required sample size
Z = Z-score corresponding to the desired level of confidence (in this case 0.075)
p = estimated population proportion (0.5 as it provides the maximum variance)
E = margin of error (0.03)
Plugging in the values, we get:
n = (Z^2 * p * (1-p))/(E^2)
n = (Z^2 * 0.5 * 0.5)/(0.03^2)
n = (1.96^2 * 0.5 * 0.5)/(0.03^2)
n = (3.8416 * 0.25)/(0.0009)
n = 0.9604/0.0009
n = 1069.33
Therefore, the sample size needed to guarantee that the requirement is met is 1070 voters.