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 $99\mathrm{\%}$ confidence if
(a) she uses a previous estimate of 0.56 ?
(b) she does not use any prior estimates?

E Click the icon to view the table of critical values.
(a) $n=1022$ (Round up to the nearest integer)
(b) $n=◻$ (Round up to the nearest integer)
Table of critical values
$$\text{Unknown environment 'tabular'}$$ & Area in Each Tail, $\frac{\mathit{\alpha }}{\mathbf{2}}$ & Critical Value, $z_1$ \
\hline $90\mathrm{\%}$ & 0.05 & 1645 \
\hline $95\mathrm{\%}$ & 0.005 & 196 \
\hline $99\mathrm{\%}$ & 0.005 & 2.575 \
\hline
\end{tabular}
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Step 1 ：The sample size needed for a proportion estimate can be calculated using the formula: $n=\frac{Z^2\cdot p\cdot (1-p)}{E^2}$, where $n$ is the sample size, $Z$ is the z-score, $p$ is the estimated proportion, and $E$ is the margin of error.

Step 2 ：For part (a), we are given a previous estimate of $p=0.56$, a margin of error $E=0.04$, and a confidence level of 99%, which corresponds to a z-score $Z=2.575$.

Step 3 ：For part (b), if no prior estimate is given, we use $p=0.5$ as this maximizes the product $p\cdot (1-p)$ and thus gives the largest possible sample size.

Step 4 ：Let's calculate the sample size for each part.

Step 5 ：For part (a), substituting the given values into the formula, we get $n_a=\frac{(2.575{)}^2\cdot 0.56\cdot (1-0.56)}{(0.04{)}^2}=1022$.

Step 6 ：For part (b), substituting the given values into the formula, we get $n_b=\frac{(2.575{)}^2\cdot 0.5\cdot (1-0.5)}{(0.04{)}^2}=1037$.

Step 7 ：Final Answer: (a) The required sample size when using a previous estimate of 0.56 is $\boxed{{\displaystyle 1022}}$. (b) The required sample size when not using any prior estimates is $\boxed{{\displaystyle 1037}}$.