The bookstore at IRSC would like to estimate the proportion of students who prefer electronic textbooks (eBooks) over printed textbooks (hard copies). A random sample of 30 students was surveyed. Their preferences are listed below.
\begin{tabular}{|c|c|c|c|c|}
\hline eBook & eBook & eBook & eBook & eBook \\
\hline eBook & eBook & eBook & eBook & eBook \\
\hline hard copy & hard copy & hard copy & hard copy & hard copy \\
\hline hard copy & hard copy & hard copy & hard copy & hard copy \\
\hline hard copy & hard copy & hard copy & hard copy & hard copy \\
\hline hard copy & hard copy & hard copy & hard copy & hard copy \\
\hline
\end{tabular}
Determine the point estimate, $\widehat{p}$ and the sample standard deviation, $s_{\widehat{p}}$. Round the sample proportion to four decimal places and round the standard deviation to six decimal places, if necessary.
\[
\begin{array}{l}
\widehat{p}=0.3333 \vee \sigma^{8} \\
s_{\widehat{p}}=0.086066 \vee \sigma^{8}
\end{array}
\]
Using a $98 \%$ confidence level, determine the margin of error, $E$, and a confidence interval for the proportion of all students at the college who work prefer eBooks over printed textbooks. Report the confidence interval using interval notation. Report the solutions in percent form, rounded to two decimal places, if necessary.
The margin of error is given by $E=20.02 \quad 0^{\circ}$.
A $98 \%$ confidence interval is given by $(13.31,53.36) \checkmark 0^{\circ}$.
The margin of error for this confidence interval is quite large. What is the best way for the bookstore to achieve a confidence interval with a smaller margin of error?
Decrease the level of confidence
Decrease the sample size
Increase the sample size
Increase the level of confidence
Answer
To achieve a confidence interval with a smaller margin of error, the bookstore should increase the sample size. Increasing the sample size will decrease the standard deviation, which in turn will decrease the margin of error.
Step 1 ：Count the number of students who prefer eBooks and those who prefer hard copies. From the table, there are 10 students who prefer eBooks and 20 who prefer hard copies.
Step 2 ：Calculate the point estimate, \(\widehat{p}\), which is the proportion of students who prefer eBooks. This is calculated as the number of students who prefer eBooks divided by the total number of students. So, \(\widehat{p} = \frac{10}{30} = 0.3333\) (rounded to four decimal places).
Step 3 ：Calculate the sample standard deviation, \(s_{\widehat{p}}\), using the formula \(\sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}}\), where \(n\) is the total number of students. So, \(s_{\widehat{p}} = \sqrt{\frac{0.3333(1-0.3333)}{30}} = 0.086066\) (rounded to six decimal places).
Step 4 ：Calculate the margin of error, \(E\), using the formula \(E = z*\sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}}\), where \(z\) is the z-score corresponding to the desired confidence level. For a 98% confidence level, the z-score is approximately 2.33. So, \(E = 2.33*\sqrt{\frac{0.3333(1-0.3333)}{30}} = 0.2002\) or 20.02% (in percent form).
Step 5 ：Calculate the confidence interval, which is \(\widehat{p} \pm E\), which is \(0.3333 \pm 0.2002\) or (13.31%, 53.36%) in percent form. \(\boxed{(13.31\%, 53.36\%)}\)
Step 6 ：To achieve a confidence interval with a smaller margin of error, the bookstore should increase the sample size. Increasing the sample size will decrease the standard deviation, which in turn will decrease the margin of error.
