Question 15 of 24 This question: 2 point(s) possible Submit test 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 0.02 with $95 \%$ confidence if (a) she uses a previous estimate of 038 ? (b) she does not use any prior estimates? E) Click the icon to view the table of critical values. (a) $n=\square$ (Round up to the nearest integer) (b) $n=\square$ (Round up to the nearest integer)

Step 1 ：We are given that the researcher wishes to estimate the proportion of adults who have high-speed Internet access. She wants the estimate to be within 0.02 with 95% confidence.

Step 2 ：For a 95% confidence level, the Z-score is 1.96.

Step 3 ：For scenario (a), she uses a previous estimate of 0.38. We can use the formula for sample size \(n = \frac{{Z^2 \cdot p \cdot (1-p)}}{{E^2}}\), where \(Z = 1.96\), \(E = 0.02\), and \(p = 0.38\).

Step 4 ：Substituting the values into the formula, we get \(n = \frac{{(1.96)^2 \cdot 0.38 \cdot (1-0.38)}}{{(0.02)^2}}\).

Step 5 ：Calculating the above expression, we get \(n \approx 2262.3\). Since the sample size must be an integer, we round up to get \(n = 2263\).

Step 6 ：For scenario (b), she does not use any prior estimates. In this case, we use \(p = 0.5\) to maximize the sample size.

Step 7 ：Substituting the values into the formula, we get \(n = \frac{{(1.96)^2 \cdot 0.5 \cdot (1-0.5)}}{{(0.02)^2}}\).

Step 8 ：Calculating the above expression, we get \(n \approx 2400.25\). Since the sample size must be an integer, we round up to get \(n = 2401\).

Step 9 ：Final Answer: (a) The sample size should be \(\boxed{2263}\). (b) The sample size should be \(\boxed{2401}\).