A sample of 20 elements produced a mean of 91.2 and a standard deviation of 12.84. Assuming
that the population has a normal distribution, the 90% confidence interval for the population
mean, rounded to two decimal places, is:


To find the 90% confidence interval for the population mean, we will use the formula for a confidence interval based on the sample mean:

\[
CI = \bar{x} \pm Z_{\alpha/2} \times \frac{s}{\sqrt{n}}
\]

Where:
- \( \bar{x} \) is the sample mean (91.2),
- \( s \) is the sample standard deviation (12.84),
- \( n \) is the sample size (20),
- \( Z_{\alpha/2} \) is the Z-value corresponding to a 90% confidence level, which is 1.645 for a two-tailed test.

### Step-by-step calculation:
1. Compute the standard error (SE):
\[
SE = \frac{s}{\sqrt{n}} = \frac{12.84}{\sqrt{20}}
\]

2. Multiply the Z-value by the standard error:
\[
Margin\ of\ Error = Z_{\alpha/2} \times SE
\]

3. Add and subtract the margin of error from the sample mean to find the confidence interval.

Let's calculate it.

The 90% confidence interval for the population mean is approximately \( (86.48, 95.92) \).

Would you like more details on how this was calculated, or do you have any further questions?

Here are 5 related questions to explore further:
1. How would the confidence interval change if the sample size were larger?
2. What is the effect of using a 95% confidence level instead of 90%?
3. How is the Z-value determined for different confidence levels?
4. How do you interpret a confidence interval in real-world scenarios?
5. What happens if the population distribution is not normal?

**Tip**: A larger sample size decreases the margin of error, leading to a narrower confidence interval.

A sample of 20 elements produced a mean of 91.2 and a standard deviation of 12.84. Assuming that the population has a normal distribution, the 90% confidence interval for the population mean, rounded to two decimal places, is:

This solution explains how to calculate the 90% confidence interval for a population mean, using a sample mean of 91.2 and a standard deviation of 12.84 from 20 elements. It covers the key statistical concepts like confidence intervals, standard error, and the normal distribution, along with step-by-step calculations.

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