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.
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Part 1
​(a) Figure the​ 95% confidence interval for samples of size 2.
One can be​ 95% confident that the true population mean is between 
  
enter your response here and 
  
enter your response here.

To calculate the 95% confidence interval for a population with a known standard deviation, we can use the formula:

\[
\text{Confidence Interval} = \bar{x} \pm z_{\alpha/2} \times \left(\frac{\sigma}{\sqrt{n}}\right)
\]

Where:
- \(\bar{x}\) = sample mean (98 in this case)
- \(z_{\alpha/2}\) = z-score corresponding to the 95% confidence level (1.96 for 95%)
- \(\sigma\) = population standard deviation (22)
- \(n\) = sample size (2 for part a)

### Step-by-step Calculation:

1. **Find the standard error (SE)**:
\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{22}{\sqrt{2}} = \frac{22}{1.414} \approx 15.556
\]

2. **Find the margin of error**:
\[
\text{Margin of Error} = z_{\alpha/2} \times SE = 1.96 \times 15.556 \approx 30.689
\]

3. **Find the confidence interval**:
\[
\text{Lower Limit} = \bar{x} - \text{Margin of Error} = 98 - 30.689 \approx 67.311
\]
\[
\text{Upper Limit} = \bar{x} + \text{Margin of Error} = 98 + 30.689 \approx 128.689
\]

### Conclusion:
One can be 95% confident that the true population mean is between **67.31** and **128.69**.

Would you like further details on this or the next parts of the question?

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Here are 5 relative questions to expand your understanding:
1. How does increasing the sample size affect the confidence interval?
2. What is the effect of a larger population standard deviation on the confidence interval?
3. How does changing the confidence level (e.g., 99%) change the width of the confidence interval?
4. What happens if the population doesn't follow a normal distribution?
5. How is the confidence interval interpreted in practical research?

**Tip:** As the sample size increases, the confidence interval becomes narrower, making your estimate 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. Part (a) Figure the 95% confidence interval for samples of size 2.

Learn how to calculate the 95% confidence interval for a sample size of 2, assuming a population standard deviation of 22 and a sample mean of 98. Follow this step-by-step guide to understanding confidence intervals and their relationship to normal distribution, z-scores, and standard error. Suitable for advanced high school or undergraduate statistics students.

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