For a population that has a standard deviation of 

10​, 

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 

100 

and that the population of individuals is known to follow a normal curve.Figure the​ 95% confidence interval for samples of size 

3.

One can be​ 95% confident that the true population mean is between 


To calculate the 95% confidence interval for a population mean, given a known standard deviation and a sample size, we use the following formula:

\[
\text{Confidence Interval} = \bar{x} \pm Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}
\]

Where:
- \(\bar{x}\) is the sample mean.
- \(Z_{\alpha/2}\) is the Z-value corresponding to the desired confidence level (for 95%, \(Z_{\alpha/2} \approx 1.96\)).
- \(\sigma\) is the population standard deviation.
- \(n\) is the sample size.

Given:
- Sample mean, \(\bar{x} = 100\)
- Population standard deviation, \(\sigma = 10\)
- Sample size, \(n = 3\)

Let's plug in the values to calculate the confidence interval:

\[
\text{Standard Error} = \frac{\sigma}{\sqrt{n}} = \frac{10}{\sqrt{3}} \approx 5.77
\]

\[
\text{Margin of Error} = Z_{\alpha/2} \times \text{Standard Error} = 1.96 \times 5.77 \approx 11.31
\]

Finally, the confidence interval is:

\[
\text{Confidence Interval} = 100 \pm 11.31
\]

So, the 95% confidence interval is:

\[
(100 - 11.31, 100 + 11.31) = (88.69, 111.31)
\]

### Answer:
One can be 95% confident that the true population mean is between **88.69** and **111.31**.

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Would you like more details or have any questions? Here are some related questions:

1. How does the sample size affect the confidence interval?
2. What is the difference between a 95% confidence interval and a 99% confidence interval?
3. How do you interpret the margin of error in a confidence interval?
4. What assumptions are necessary for the confidence interval to be valid?
5. How would the confidence interval change if the standard deviation were unknown?

**Tip:** When interpreting confidence intervals, remember that they provide a range of plausible values for the population mean, not a guarantee that the mean falls within that range.

For a population that has a standard deviation of 10, 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 100 and that the population of individuals is known to follow a normal curve. Figure the 95% confidence interval for samples of size 3. One can be 95% confident that the true population mean is between

Learn how to calculate the 95% confidence interval for a sample with a mean of 100, a standard deviation of 10, and a sample size of 3. This guide covers key statistical concepts, such as confidence intervals, the Z-value for 95% confidence, and the margin of error. Suitable for undergraduate statistics students.

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