1. **Problem Statement:** We have a population with values $1, 2, 8, 9$ and sample size $n=2$ drawn without replacement. 2. **List all possible samples of size 2 without replacement:** $\{1,2\}, \{1,8\}, \{1,9\}, \{2,8\}, \{2,9\}, \{8,9\}$ 3. **Calculate the population mean $\mu$:** $$\mu = \frac{1+2+8+9}{4} = \frac{20}{4} = 5$$ 4. **Calculate the population variance $\sigma^2$:** $$\sigma^2 = \frac{(1-5)^2 + (2-5)^2 + (8-5)^2 + (9-5)^2}{4} = \frac{16 + 9 + 9 + 16}{4} = \frac{50}{4} = 12.5$$ 5. **Calculate the population standard deviation $\sigma$:** $$\sigma = \sqrt{12.5} = 3.5355$$ 6. **Calculate the mean of the sampling distribution of sample means $\mu_{\bar{x}}$:** The mean of sample means equals the population mean: $$\mu_{\bar{x}} = \mu = 5$$ 7. **Calculate the variance of the sampling distribution of sample means $\sigma^2_{\bar{x}}$:** Using the formula for sampling without replacement: $$\sigma^2_{\bar{x}} = \frac{\sigma^2}{n} \times \frac{N - n}{N - 1} = \frac{12.5}{2} \times \frac{4 - 2}{4 - 1} = 6.25 \times \frac{2}{3} = 4.1667$$ 8. **Calculate the standard deviation of the sampling distribution of sample means $\sigma_{\bar{x}}$:** $$\sigma_{\bar{x}} = \sqrt{4.1667} = 2.0412$$