For a population that has a standard deviation of 

22,

figure the​ 95% confidence interval​ (that is, the lower and upper confidence​ limits) for parts​ (a) through​ (d). Assume that in each case the​ researcher's sample has a mean of 

98,

and that the population of individuals is known to follow a normal curve.

To compute the **95% confidence interval** for a population with a known standard deviation, we can use the formula for the confidence interval of the population mean when the standard deviation is known. This is typically done using a **Z-distribution**.

The formula for the confidence interval is:

\[
\text{CI} = \bar{X} \pm Z_{\alpha/2} \times \left(\frac{\sigma}{\sqrt{n}}\right)
\]

Where:
- \(\bar{X}\) is the sample mean (which is given as 98).
- \(Z_{\alpha/2}\) is the Z-score corresponding to the confidence level (for 95% confidence, \(Z_{\alpha/2} = 1.96\)).
- \(\sigma\) is the population standard deviation (given as 22).
- \(n\) is the sample size.

To compute the confidence interval, we need to know the sample sizes for each part. Could you provide those sample sizes for parts (a) through (d), or would you like me to assume some sample sizes for demonstration?

Let me know if you have any questions or need further details.

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### Five related questions:
1. How does the sample size affect the width of a confidence interval?
2. Why do we use the Z-distribution instead of the T-distribution in this case?
3. What happens to the confidence interval if the confidence level increases to 99%?
4. How does the confidence interval change if the population standard deviation is not known?
5. Why is a normal distribution assumption important for calculating confidence intervals?

### Tip:
Remember that a larger sample size will result in a narrower confidence interval, making the estimate of the population mean more precise.

For a population that has a standard deviation of 22, figure the 95% confidence interval (that is, the lower and upper confidence limits) for parts (a) through (d). Assume that in each case the researcher's sample has a mean of 98, and that the population of individuals is known to follow a normal curve.

Learn how to compute a 95% confidence interval for a population with a standard deviation of 22. This step-by-step guide covers key statistical concepts such as the Z-distribution, confidence intervals, and normal distribution assumptions. Suitable for college-level statistics students.

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