Question
Suppose lengths of text messages are normally distributed and have a known population standard deviation of 3 characters and an unknown population mean. A random sample of 22 text messages is taken and gives a sample mean of 31 characters.
Identify the parameters needed to calculate a confidence interval at the $90 \%$ confidence level. Then find the confidence interval.
\begin{tabular}{ccccc}
$\mathrm{z}_{0.10}$ & $\mathrm{z}_{0.05}$ & $\mathrm{z}_{0.025}$ & $\mathrm{z}_{0.01}$ & $\mathrm{z}_{0.005}$ \\
\hline 1.282 & 1.645 & 1.960 & 2.326 & 2.576
\end{tabular}
Use the common $\mathrm{z}$ values given above and enter as shown above.
- Round the final answer to two decimal places, if necessary.

Step 1 ：Given that the lengths of text messages are normally distributed with a known population standard deviation of 3 characters and an unknown population mean. A random sample of 22 text messages gives a sample mean of 31 characters.

Step 2 ：We are asked to calculate a confidence interval at the 90% confidence level. The parameters needed for this calculation are the sample mean, the standard deviation, the sample size, and the z-score corresponding to the desired confidence level.

Step 3 ：In this case, we have a sample mean of 31 characters, a standard deviation of 3 characters, a sample size of 22 text messages, and we want a 90% confidence level.

Step 4 ：The z-score for a 90% confidence level is not given directly in the table, but we can calculate it as follows: since the total area under the normal curve is 1, and we want the middle 90% for our confidence interval, this leaves 10% in the two tails of the distribution. Since the normal distribution is symmetric, this means there is 5% in each tail, so we should use the z-score for 0.05, which is 1.645.

Step 5 ：The formula for a confidence interval is: \[\bar{x} \pm z \frac{\sigma}{\sqrt{n}}\] where \(\bar{x}\) is the sample mean, \(z\) is the z-score, \(\sigma\) is the standard deviation, and \(n\) is the sample size.

Step 6 ：Let's plug in the values and calculate the confidence interval. After running this code, we will observe the lower and upper bounds of the confidence interval.

Step 7 ：Final Answer: The 90% confidence interval for the population mean length of text messages is \(\boxed{(29.95, 32.05)}\) characters.