To calculate the confidence intervals for the population mean $\mu$ using the sample information provided, we can use the following formula for the confidence interval when the population standard deviation $\sigma$ is known:

$\text{Confidence Interval} = \left( \bar{x} - Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}, \bar{x} + Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} \right)$

Where:

• $\bar{x} = 40$ (sample mean)
• $\sigma = 7$ (population standard deviation)
• $n = 13$ (sample size)
• $Z_{\alpha/2}$ is the critical value from the standard normal distribution for the desired confidence level.

Step-by-Step Calculations:

1. Calculate the Standard Error (SE): $SE = \frac{\sigma}{\sqrt{n}} = \frac{7}{\sqrt{13}} \approx 1.940$

2. Determine the critical value $Z_{\alpha/2}$ for different confidence levels:

   • For a 90% confidence level, $Z_{\alpha/2} \approx 1.645$
   • For a 95% confidence level, $Z_{\alpha/2} \approx 1.96$
   • For a 99% confidence level, $Z_{\alpha/2} \approx 2.576$

3. Calculate the confidence intervals:

   • 90% Confidence Interval: $40 \pm 1.645 \times 1.940 = 40 \pm 3.189$ \text{90% CI} = (36.811, 43.189)

   • 95% Confidence Interval: $40 \pm 1.96 \times 1.940 = 40 \pm 3.802$ \text{95% CI} = (36.198, 43.802)

   • 99% Confidence Interval: $40 \pm 2.576 \times 1.940 = 40 \pm 4.998$ \text{99% CI} = (35.002, 44.998)

Summary:

• 90% CI: $(36.811, 43.189)$
• 95% CI: $(36.198, 43.802)$
• 99% CI: $(35.002, 44.998)$

Would you like more details on any of these calculations, or do you have any other questions?

Follow-up Questions:

1. What are the implications of using different confidence levels?
2. How does sample size affect the width of a confidence interval?
3. What is the difference between using a Z-distribution and a T-distribution in confidence intervals?
4. How would the confidence intervals change if the population standard deviation were not known?
5. What is the relationship between confidence level and the margin of error?
6. How can confidence intervals be used in hypothesis testing?
7. What happens to the confidence interval if the sample mean changes?
8. Why is it important to assume that the sample is from a normal population?

Tip:

Always ensure that the conditions for using a Z-distribution are met, such as knowing the population standard deviation and having a normally distributed population or a large sample size.