https://file.aiquickdraw.com/mathbot/file/1728982821733l9uv54uv.jpg

Let's work through the question from the image. Here's a breakdown of the problem:

**Given:**
- Sample size \(n = 25\)
- Population size \(N = 600\)
- Population mean \(\mu = 108.4\)
- Population variance \(\sigma^2 = 812.25\)

### 1. **Expected Value of the Sampling Distribution:**
The expected value (mean) of the sampling distribution of the sample mean is equal to the population mean:

\[
E(\bar{X}) = \mu = 108.4
\]

### 2. **Standard Error of the Sampling Distribution:**
Since sampling is done **without replacement**, we need to apply the finite population correction factor (FPC) to adjust the standard error. 

The standard error (SE) of the sampling distribution of the sample mean is given by:

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

Where:
- \(\sigma = \sqrt{812.25} = 28.5\) (standard deviation of the population)
- \(n = 25\) (sample size)
- \(N = 600\) (population size)

Now, plugging in the values:

\[
SE(\bar{X}) = \frac{28.5}{\sqrt{25}} \times \sqrt{\frac{600 - 25}{600 - 1}} = \frac{28.5}{5} \times \sqrt{\frac{575}{599}}
\]

Simplifying:

\[
SE(\bar{X}) = 5.7 \times \sqrt{0.960}
\]

\[
SE(\bar{X}) \approx 5.7 \times 0.98 = 5.59
\]

### **Final Answers:**
- **Expected value** of the sampling distribution: \( E(\bar{X}) = 108.4 \)
- **Standard error** of the sampling distribution: \( SE(\bar{X}) \approx 5.59 \)

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

### Related Questions:
1. How would the expected value change if the population mean were different?
2. What happens to the standard error if sampling is done **with replacement**?
3. How does the finite population correction factor influence the standard error?
4. What effect does increasing the sample size have on the standard error?
5. How would the calculation differ if the population size were infinite?

### Tip:
When sampling from a finite population without replacement, always remember to apply the finite population correction factor to get a more accurate standard error.

Samples of size 25 are taken from a population of 600 values. The population has a mean of 108.4 and a variance equal to 812.25. Obtain the expected value and standard error of the sampling distribution of means if sampling is done without replacement.

This problem involves calculating the expected value and standard error for a sampling distribution of means, with a population mean of 108.4 and variance 812.25. The finite population correction factor is applied since sampling is done without replacement.

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