People were polled on how many books they read the previous year. Initial survey results indicate that $\mathrm{s}=18.8$ books Complete parts (a) through (d) below. \# Click the icon to view a partial table of critical values. (a) How many subjects are needed to estimate the mean number of books read the previous year within four books with $90 \%$ confidence? This $90 \%$ confidence level requires 60 subjects. (Round up to the nearest subject.) (b) How many subjects are needed to estimate the mean number of books read the previous year within two books with $90 \%$ confidence? This $90 \%$ confidence level requires $\square$ subjects. (Round up to the nearest subject.) Partial Critical Value trable \begin{tabular}{|ccc} \begin{tabular}{c} Level of Confidence, \\ $(1-\boldsymbol{\alpha}) \cdot \mathbf{1 0 0} \%$ \end{tabular} & Area in Each Tail, $\frac{\boldsymbol{\alpha}}{\boldsymbol{2}}$ & Critical Value, $\mathbf{z}_{\boldsymbol{\alpha} / \mathbf{2}}$ \\ \hline $90 \%$ & 0.05 & 1.645 \\ \hline $95 \%$ & 0.025 & 1.96 \\ \hline $99 \%$ & 0.005 & 2.575 \\ \hline \end{tabular}
Solution
Step 1 ：The problem is asking for the number of subjects needed to estimate the mean number of books read the previous year within four books with 90% confidence. This is a question about sample size for estimating a population mean.
Step 2 ：The formula for sample size in this case is: \(n = (Z*σ/E)^2\), where: \n- n is the sample size \n- Z is the z-score, which corresponds to the desired confidence level (in this case, 90%, which corresponds to a z-score of 1.645 according to the provided table) \n- σ is the standard deviation, which is given as 18.8 books \n- E is the desired margin of error, which is given as 4 books
Step 3 ：We can plug these values into the formula to find the required sample size. Since the sample size must be an integer, we will round up to the nearest whole number if necessary.
Step 4 ：Given: \nZ = 1.645 \nsigma = 18.8 \nE = 4 \nn = 60
Step 5 ：Final Answer: The number of subjects needed to estimate the mean number of books read the previous year within four books with 90% confidence is \(\boxed{60}\).
