Let's solve the problem step by step.

Information provided:

• Confidence level ($c$) = 0.95
• Sample mean ($\bar{x}$) = 39.1
• Population standard deviation ($\sigma$) = 8.7
• Population size ($N$) = 400
• Sample size ($n$) = 36

Step 1: Find the finite population correction factor

The formula for the finite population correction factor is:

$\sqrt{\frac{N - n}{N - 1}}$

Substituting the values: $\sqrt{\frac{400 - 36}{400 - 1}} = \sqrt{\frac{364}{399}} \approx \sqrt{0.912} \approx 0.955$

Step 2: Find the standard error of the mean

The formula for the standard error of the mean adjusted for finite population is:

$\sigma_{\bar{x}} = \sigma \cdot \frac{1}{\sqrt{n}} \cdot \sqrt{\frac{N - n}{N - 1}}$

Substituting the values: $\sigma_{\bar{x}} = 8.7 \cdot \frac{1}{\sqrt{36}} \cdot 0.955$

$\sigma_{\bar{x}} = 8.7 \cdot 0.1667 \cdot 0.955 \approx 1.384$

Step 3: Find the critical value ($z$) for 95% confidence level

From the standard normal distribution table, the critical value for $c = 0.95$ is:

$z = 1.96$

Step 4: Calculate the margin of error

The margin of error is given by:

$E = z \cdot \sigma_{\bar{x}}$

Substituting the values: $E = 1.96 \cdot 1.384 \approx 2.71$

Step 5: Construct the confidence interval

The confidence interval is:

$\bar{x} \pm E$

Substituting the values: $39.1 \pm 2.71$

$\text{Lower bound: } 39.1 - 2.71 = 36.39$ $\text{Upper bound: } 39.1 + 2.71 = 41.81$

Final Answer:

The 95% confidence interval for the population mean is:

$(36.39, 41.81)$

Let me know if you have any questions or need further clarification!

Related Questions:

1. How does the finite population correction factor influence the confidence interval calculation?
2. What is the significance of using $z = 1.96$ for a 95% confidence level?
3. How would the interval change if the sample size were larger?
4. Why is the standard deviation of the population used instead of the sample standard deviation in this scenario?
5. What happens to the confidence interval if the population size $N$ increases significantly?

Tip:

Always check if the sample size is sufficiently large or small compared to the population to decide whether the finite population correction factor is needed.