Questions: Module Overview and Activities - MTH-102-0 10.2 Confidence Interval for Mean - Population Standard Deviation Known The pages per book in a library are normally distributed with a population standard deviation of 33 pages and an unknown population mean. A random sample of 16 books is taken and results in a sample mean of 334 pages. Identify the parameters needed to calculate a confidence interval at the 99% confidence level. Then find the confidence interval. z0.10 z0.05 z0.025 z0.01 z0.005 1.282 1.645 1.960 2.326 2.576 Use the common z values given above. - Round the final confidence interval endpoints to the nearest hundredths place. Provide your answer below: barx= sigma= n= zfraca2=

$Module Overview and Activities - MTH-102-0 10.2 Confidence Interval for Mean - Population Standard Deviation Known The pages per book in a library are normally distributed with a population standard deviation of 33 pages and an unknown population mean. A random sample of 16 books is taken and results in a sample mean of 334 pages. Identify the parameters needed to calculate a confidence interval at the 99% confidence level. Then find the confidence interval. z0.10 z0.05 z0.025 z0.01 z0.005 1.282 1.645 1.960 2.326 2.576 Use the common z values given above. - Round the final confidence interval endpoints to the nearest hundredths place. Provide your answer below: barx= sigma= n= zfraca2=$

Transcript text: Module Overview and Activities - MTH-102-0 10.2 Confidence Interval for Mean - Population Standard Deviation Known The pages per book in a library are normally distributed with a population standard deviation of 33 pages and an unknown population mean. A random sample of 16 books is taken and results in a sample mean of 334 pages. Identify the parameters needed to calculate a confidence interval at the $99 \%$ confidence level. Then find the confidence interval. \begin{tabular}{ccccc} $z_{0.10}$ & $z_{0.05}$ & $z_{0.025}$ & $z_{0.01}$ & $z_{0.005}$ \\ \hline 1.282 & 1.645 & 1.960 & 2.326 & 2.576 \\ \hline \end{tabular} Use the common $z$ values given above. - Round the final confidence interval endpoints to the nearest hundredths place. Provide your answer below: \[ \begin{array}{c} \bar{x}=\square \\ \sigma=\square \\ n=\square \\ z_{\frac{a}{2}}=\square \\ \square \end{array} \]

Solution

Solution Steps

Step 1: Identify Parameters

We are given the following parameters for the confidence interval calculation:

Sample Mean: $\bar{x} = 334$
Population Standard Deviation: $\sigma = 33$
Sample Size: $n = 16$
Confidence Level: $99\%$ (which corresponds to a significance level $\alpha = 0.01$)

Step 2: Determine the Z-score

For a $99\%$ confidence level, the critical value (Z-score) corresponding to $\frac{\alpha}{2} = 0.005$ is: \[ z_{\frac{\alpha}{2}} = z_{0.005} = 2.576 \]

Step 3: Calculate the Margin of Error

The margin of error (ME) is calculated using the formula: \[ ME = z_{\frac{\alpha}{2}} \cdot \frac{\sigma}{\sqrt{n}} \] Substituting the known values: \[ ME = 2.576 \cdot \frac{33}{\sqrt{16}} = 2.576 \cdot \frac{33}{4} = 2.576 \cdot 8.25 = 21.25 \]

Step 4: Calculate the Confidence Interval

The confidence interval is given by: \[ \bar{x} \pm ME \] Thus, we have: \[ \text{Lower Bound} = 334 - 21.25 = 312.75 \] \[ \text{Upper Bound} = 334 + 21.25 = 355.25 \] Therefore, the confidence interval is: \[ (312.75, 355.25) \]