A researcher wishes to estimate the proportion of adults who have high-speed Internet access. What size sample should be obtained if she wishes the estimate to be within 0.04 with 90% confidence if
(a) she uses a previous estimate of 0.34?
(b) she does not use any prior estimates?
a. n=?
b.n = ?
a. Using a previous estimate of 0.34, we can use the formula:
n = (z^2 * p * (1-p)) / E^2
where:
z = the z-score corresponding to the desired level of confidence (in this case, 90% corresponds to a z-score of 1.645)
p = the previous estimate (0.34)
E = the margin of error (0.04)
Plugging in these values, we get:
n = (1.645^2 * 0.34 * (1-0.34)) / 0.04^2
n = 422.23
Round up to the nearest whole number to get a sample size of 423.
b. Without any prior estimate, we can use a conservative estimate of p = 0.5 (since this is the most uncertain estimate and will result in the largest sample size). Using the same formula as above, we get:
n = (1.645^2 * 0.5 * (1-0.5)) / 0.04^2
n = 601.06
Round up to the nearest whole number to get a sample size of 602.