PLEASE HELP ME
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 p̂ = 0.34 and a margin of error of E = 0.04, we can use the formula:
n = [(zα/2)² * p̂ * (1 - p̂)] / E²
where zα/2 is the z-score corresponding to the level of confidence (90% in this case), and p̂ is the previous estimate.
Plugging in the values, we get:
n = [(1.645)² * 0.34 * (1 - 0.34)] / 0.04² ≈ 647.53
Rounding up to the nearest whole number, we need a sample size of n = 648.
b. Without any prior estimate, we can use a conservative estimate of p̂ = 0.5 (since we don't know the true proportion), and the same margin of error of E = 0.04. Using the same formula as before, we get:
n = [(1.645)² * 0.5 * (1 - 0.5)] / 0.04² ≈ 601.04
Rounding up, we need a sample size of n = 602.