Section 10.7 Exercises
Given Y = 30 successes in n = 100 trials, determine a 90% confidence interval for the unknown value for p.
Solution.
\(\tilde{p} = 0.3\) and \(z_{0.05} = 1.645\) gives
\begin{equation*}
0.3 - 1.645 \sqrt{\frac{0.3 \cdot 0.7}{100}} \lt p \lt 0.3 + 1.645 \sqrt{\frac{0.3 \cdot 0.7}{100}}
\end{equation*}
or
\begin{equation*}
0.225 \lt p \lt 0.375.
\end{equation*}
Checkpoint 10.7.2. - Sample Size for confidence interval for p.
Given a preliminary estimate \(\tilde{p_0} = 0.23\text{,}\) determine the same size needed for determine a 95% confidence interval for p with margin of error 0.02.
Solution.Using \(z_{0.025} = 1.96\text{,}\)
\begin{equation*}
n \gt \big ( \frac{1.96}{0.02} \big )^2 \cdot 0.23 \cdot 0.77 \approx 1700.87
\end{equation*}
and so pick at least 1701 as the sample size.
Checkpoint 10.7.3. - Voting projection.
Randomly polling 3200 eligible voters for governor in a particular state resulted in finding that 1590 favored your candidate. Determine an appropriate 95% confidence interval for the true proportion p of voters who favor your candidate. Noting that \(\tilde{p}\) in this instance is smaller than 50%, write a short paragraph regarding what you might conclude from this confidence interval regarding your candidate’s chances in winning the election.
Solution.Note that although the point estimate is below 50%, the confidence interval includes the possibility that the actual value for p is greater than 50%. So, you cannot conclude that your candidate will either win or lose.
Checkpoint 10.7.4. - Basic Confidence interval for the mean.
Given a sample mean of \(\overline{x} = 25.3\) with n = 121 and sample variance \(s^2 = 12.1\text{,}\) determine a 99% confidence interval for the true mean \(\mu\text{.}\)
Solution.Using \(z_{0.005} = 2.576\) and \(s = \sqrt{12.1} \approx 3.4786\) gives a confidence interval
\begin{equation*}
25.3 - 2.576 \cdot \frac{3.4786}{11} \lt \mu \lt 25.3 + 2.576 \cdot \frac{3.4786}{11}
\end{equation*}
or
\begin{equation*}
24.4854 \lt \mu \lt 26.1148.
\end{equation*}
Checkpoint 10.7.5. - Confidence Interval Experiment.
TBA
Roll two regular pair of dice 35 times, recording the sum of the dots for each roll. Using the data from your sample, determine the corresponding sample mean and sample variance. Using this data, create a 95% confidence interval for the population mean and a 95% "centered" confidence interval for the standard deviation. Once complete, compare your results with what you know should be the correct population statistics an appropriate "hat" distribution. (You might want to use the chi-square calculator provided earlier in this text.)
Go back over your 35 rolls and count the number of 7’s or 11’s rolled. Determine a corresponding relative frequency for this outcome. Using this data, create a 95% confidence interval for the theoretical proportion of success p. Compare your result with what you know should be the correct theoretical p.
Repeat this exercise but this time roll 105 times. Notice how these differ from the confidence intervals created with the smaller set. Write a paragraph describing how these compare and whether one is better or not than the other.
