The problem presented in the image involves calculating the expected value and the standard error of the sampling distribution of means, given specific population parameters. Here's how we solve it:

Problem Breakdown:

• Population size (N) = 600
• Sample size (n) = 25
• Population mean (μ) = 108.4
• Population variance (σ²) = 812.25 (Population standard deviation: $\sigma = \sqrt{812.25} = 28.5$)
• Sampling is without replacement, so we need to adjust the standard error using the finite population correction factor.

1. Expected Value of the Sampling Distribution:

The expected value (mean) of the sampling distribution of sample means is the same as the population mean:

$E(\bar{X}) = \mu = 108.4$

2. Standard Error of the Sampling Distribution (with finite population correction):

The formula for the standard error of the mean, when sampling without replacement, is:

$SE(\bar{X}) = \frac{\sigma}{\sqrt{n}} \times \sqrt{\frac{N - n}{N - 1}}$

Where:

• $\sigma = 28.5$ (Population standard deviation)
• $n = 25$ (Sample size)
• $N = 600$ (Population size)

First, calculate the standard error without the finite population correction:

$SE_{\text{without correction}} = \frac{28.5}{\sqrt{25}} = \frac{28.5}{5} = 5.7$

Now, apply the finite population correction factor:

$\text{Correction factor} = \sqrt{\frac{600 - 25}{600 - 1}} = \sqrt{\frac{575}{599}} = \sqrt{0.960} \approx 0.98$

Finally, the standard error with the finite population correction is:

$SE(\bar{X}) = 5.7 \times 0.98 = 5.59$

Final Answers:

• Expected value of the sampling distribution: $E(\bar{X}) = 108.4$
• Standard error of the sampling distribution: $SE(\bar{X}) = 5.59$

Would you like a more detailed explanation or have any questions?

Related Questions:

1. How does the finite population correction affect the standard error?
2. What happens to the standard error if the sample size increases?
3. How would the answer change if sampling were done with replacement?
4. Why does the expected value of the sample mean equal the population mean?
5. What is the effect of a larger population size on the standard error?

Tip:

Always check whether sampling is done with or without replacement, as it affects the standard error calculation.