1. **State the problem:** Dmitry suspects the true proportion of odd numbers rolled is different from 0.5. 2. **Define hypotheses:** - Null hypothesis: $H_0: p = 0.5$ - Alternative hypothesis: $H_a: p \neq 0.5$ (two-tailed test) 3. **Gather data:** - Number of rolls, $n = 40$ - Number of odd outcomes, $x = 14$ 4. **Calculate sample proportion:** $$\hat{p} = \frac{x}{n} = \frac{14}{40} = 0.35$$ 5. **Calculate test statistic $z$:** Under $H_0$, the standard error is $$SE = \sqrt{\frac{p_0(1-p_0)}{n}} = \sqrt{\frac{0.5 \times 0.5}{40}} = \sqrt{\frac{0.25}{40}} = \sqrt{0.00625} = 0.07906$$ The test statistic is $$z = \frac{\hat{p} - p_0}{SE} = \frac{0.35 - 0.5}{0.07906} = \frac{-0.15}{0.07906} \approx -1.897$$ 6. **Find the p-value:** Since the test is two-tailed, the p-value is $$p = 2 \times P(Z \leq -1.897)$$ Consulting standard normal tables or Excel's NORMSDIST: $$p \approx 2 \times 0.0289 = 0.0578$$ 7. **Conclusion:** At significance level 0.05, $p = 0.0578 > 0.05$, so we do not reject $H_0$. There is not enough evidence to conclude the true proportion of odd numbers is different from 0.5. **Final answers:** Test statistic $z = -1.897$ $p$-value $= 0.058$