1. **State the problem:** We want to test if there is enough evidence to reject the claim that the average price of a home sold in Beaver County is $60,000 at a significance level of $\alpha = 0.05$. 2. **Set up hypotheses:** - Null hypothesis $H_0$: $\mu = 60000$ - Alternative hypothesis $H_a$: $\mu \neq 60000$ (two-tailed test) 3. **Calculate sample mean $\bar{x}$ and sample standard deviation $s$:** Given the 36 home prices, calculate: $$\bar{x} = \frac{\sum x_i}{n}$$ $$s = \sqrt{\frac{1}{n-1} \sum (x_i - \bar{x})^2}$$ 4. **Calculate the test statistic $t$:** $$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$$ where $\mu_0 = 60000$, $n=36$. 5. **Find critical value:** Degrees of freedom $df = n-1 = 35$. For $\alpha=0.05$ two-tailed, critical $t$ value $\approx \pm 2.030$. 6. **Decision rule:** - If $|t| > 2.030$, reject $H_0$. - Otherwise, do not reject $H_0$. 7. **Calculate sample mean:** Sum of prices = 2,022,490 $$\bar{x} = \frac{2022490}{36} \approx 56180.28$$ 8. **Calculate sample standard deviation $s$:** Calculate each $(x_i - \bar{x})^2$, sum them, divide by 35, then take square root. Sum of squared deviations $\approx 3.44 \times 10^{10}$ $$s = \sqrt{\frac{3.44 \times 10^{10}}{35}} \approx 31368.5$$ 9. **Calculate test statistic:** $$t = \frac{56180.28 - 60000}{31368.5 / \sqrt{36}} = \frac{-3819.72}{5228.08} \approx -0.73$$ 10. **Compare with critical value:** $|t| = 0.73 < 2.030$, so we do not reject $H_0$. **Final conclusion:** There is not enough evidence at the 0.05 significance level to reject the agent's claim that the average home price is $60,000$.