Question

A columnist wants to determine the proportion of columnists who support fracking at the New York Times.

1. What is the minimum sample size that needs to be collected to guarantee the 95% confidence interval to have a margin of error 0.08 for the population proportion of columnists who support fracking at the NYT'S?
2. Suppose 95 of 250 columnists supported fracking at the NYT's. Construct a 95% confidence interval for the population proportion of columnists who support fracking at the NYT's.

Answer 1

Solution-

Given that

margin of error is 0.08.

95 of 250 columnists supported fracking.

Confidence level is 95% = alpha = 0.05

The following information has been provided: a = 0.05, E = 0.08 Since the no estimate of the population proportion p is provided, we use the estimate p = 0.5 (which corresponds to the worst case scenario). The critical value for the significance level a = 0.05 is ze = 1.96. The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error: n 2p(1–p) (?) * = 0.5(1 – 0.5) ( 1:06) = 150.06 Therefore, the sample size needed to satisfy the condition n > 150.06, and it must be an integer number, we conclude that the minimum required sample size is n = 151.

95% Confidence Interval-

We need to construct the 95% confidence interval for the population proportion. We have been provided with the following information about the number of favorable cases: Favorable Cases X = 95 Sample Size N = 250 The sample proportion is computed as follows, based on the sample size N = 250 and the number of favorable cases X = 95: X 95 P = N = 250 = 0.38 The critical value for a = 0.05 is ze = 21-a/2 = 1.96. The corresponding confidence interval is computed as shown below: CH(Proportion) = (3--2009.B), + 2/10, – ) 0.38(1 – 0.38), 0.38 + 1.96 x V 0.58 250) = (0.38 – 1.96 x V 250 0.38(1 – 0.38) 250 = (0.32, 0.44) Therefore, based on the data provided, the 95% confidence interval for the population proportion is 0.32 < p < 0.44, which indicates that we are 95% confident that the true population proportion p is contained by the interval (0.32,0.44).