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+\parindent=0pt
+
+Statistical Methods
+
+\bigskip
+1a.\par
+$H_0$: Unemployment is 9\%.\par
+$H_1$: Unemployment varies significantly from 9\%.
+
+\bigskip
+1b.\par
+$H_0$: $\mu_0=55.50$\par
+$H_1$: $\mu<\mu_0$
+
+\bigskip
+1c. $H_0$: $\mu_0=45$, $H_1$: $\mu>\mu_0$.
+
+\bigskip
+1d.
+Null hypothesis: 40\% of the population gets the flu.
+Alternative hypothesis: People who take vitamin C are less likely to
+get the flu.
+
+\bigskip
+1e.
+Null hypothesis: A toothpaste manufacturer has 35\% of the
+market in Springfield.
+Alternative hypothesis:
+The toothpaste manufacturer has more than 35\% of the market in
+Springfield.
+
+\bigskip
+1f.
+Null hypothesis:
+The average rate of return on corporate A-rated bonds is 8\%.
+Alternative hypothesis:
+The average rate of return on corporate A-rated bonds is less than 8\%.
+
+\bigskip
+1g.
+Null hypothesis:
+The average number of items purchased is 14{.}56.
+Alternative hypothesis:
+The average number of items purchased differs significantly from 14{.}56.
+
+\bigskip
+1h.
+Null hypothesis:
+The average increase in money supply is 5\%.
+Alternative hypothesis:
+The average increase in money supply is greater than 5\%.
+
+\bigskip
+1i.
+Null hypothesis:
+Average delivery time is 65 days.
+Alternative hypothesis:
+Average delivery times is less than 65 days.
+
+\bigskip
+1j.
+Null hypothesis:
+Bank customers live 2 miles away on average.
+Alternative hypothesis:
+Bank customers do not live 2 miles away on average.
+
+\vfill
+\eject
+
+2. Known population variance.
+$$H_0: \mu=\mu_0$$
+$$H_1: \mu>\mu_0$$
+$$\mu_0=15$$
+$$\sigma=1.5$$
+$$n=64$$
+$$\bar Y=16.4$$
+$$\alpha=0.025$$
+$$Z^*=\sqrt n(\bar Y-\mu_0)/\sigma=7.4667$$
+$$Z_\alpha=Z_{0.025}=1.96$$
+$$7.4667\in(1.96,+\infty)$$
+Reject $H_0$ at $0.025$ level of significance.
+There is sufficient evidence that the workers are slower.
+
+\bigskip
+3. Known population variance.
+$$H_0: \mu=\mu_0$$
+$$H_1: \mu>\mu_0$$
+$$\mu_0=5.5$$
+$$\sigma=2.4$$
+$$n=64$$
+$$\bar Y=5.8$$
+$$\alpha=0.05$$
+$$Z^*=\sqrt n(\bar Y-\mu_0)/\sigma=1$$
+$$Z_\alpha=Z_{0.05}=1.645$$
+$$1\not\in(1.645,+\infty)$$
+Fail to reject $H_0$ at $0.05$ level of significance.
+There is insufficient evidence against the claim of $5.5$ miles.
+
+\vfill
+\eject
+
+4. Known population variance.
+$$H_0: \mu=\mu_0$$
+$$H_1: \mu>\mu_0$$
+$$\mu_0=1510$$
+$$\sigma=175$$
+$$n=81$$
+$$\bar Y=1560$$
+$$\alpha=0.1$$
+$$Z^*=\sqrt n(\bar Y-\mu_0)/\sigma=2.5714$$
+$$Z_\alpha=Z_{0.1}=1.282$$
+$$2.5714\in(1.282,+\infty)$$
+Reject $H_0$ at $0.1$ level of significance.
+There is sufficient evidence that the economist is correct.
+
+\bigskip
+5. Known population variance.
+$$H_0: \mu=\mu_0$$
+$$H_1: \mu>\mu_0$$
+$$\mu_0=22500$$
+$$\sigma=1200$$
+$$n=36$$
+$$\bar Y=22900$$
+$$\alpha=0.01$$
+$$Z^*=\sqrt n(\bar Y-\mu_0)/\sigma=2$$
+$$Z_\alpha=Z_{0.01}=2.326$$
+$$2\not\in(2.326,+\infty)$$
+Fail to reject $H_0$ at $0.01$ level of significance.
+We cannot conclude that UIS Mathematical Sciences
+majors start at a higher salary.
+
+\vfill
+\eject
+
+6. Unknown population variance, normal distribution.
+$$H_0:\mu=\mu_0$$
+$$H_1:\mu<\mu_0$$
+$$\mu_0=5$$
+$$n=9$$
+$$\bar Y=5.5889$$
+$$S=0.7219$$
+$$\alpha=0.05$$
+$$t^*=\sqrt n(\bar Y-\mu_0)/S=2.4473$$
+$$t_{(n-1,\alpha)}=T_{(8,0.05)}=$$
+
+\bigskip
+7. Unknown population variance, normal distribution.
+$$H_0:\mu=\mu_0$$
+$$H_1:\mu<\mu_0$$
+$$\mu_0=65$$
+$$n=16$$
+$$\bar Y=65.97$$
+$$S=4.106$$
+$$\alpha=0.01$$
+$$t^*=\sqrt n(\bar Y-\mu_0)/S=0.9483$$
+$$T_{(n-1,\alpha)}=t_{(15,0.01)}=2.603$$
+$$0.9483\not\in(-\infty,-2.603)$$
+Fail to reject $H_0$ at $0.01$ level of significance.
+The results support the supplier's claim.
+
+\vfill
+\eject
+8. Unknown population variance, normal distribution.
+$$H_0:\mu=\mu_0$$
+$$H_1:\mu<\mu_0$$
+$$\mu_0=68$$
+$$n=16$$
+$$\bar Y=65.25$$
+$$S=2.5$$
+$$\alpha=0.03$$
+$$t^*=\sqrt n(\bar Y-\mu_0)/S=$$
+
+\vfill
+\eject
+
+9. Hypothesis of population proportion.
+$$H_0:p=p_0$$
+$$H_1:p\ne p_0$$
+$$p_0=0.5$$
+$$n=225$$
+$$Y=130$$
+$$\alpha=0.05$$
+$$\hat p=Y/n=$$
+$$Z^*={\hat p - p_0\over\sqrt{p_0(1-p_0)/n}}=$$
+$$Z_{\alpha/2}=Z_{0.025}=$$
+
+\vfill
+\eject
+
+10. Known population variance.
+$$H_0:\mu=\mu_0$$
+$$H_1:\mu<\mu_0$$
+$$\mu_0=32$$
+$$\sigma=5$$
+$$n=100$$
+$$\bar Y=31.34$$
+$$\alpha=0.05$$
+$$Z^*=\sqrt n(\bar Y-\mu_0)/\sigma=$$
+$$Z_\alpha=Z_{0.05}=$$
+
+\vfill
+\eject
+
+11a. Unknown population variance, normal distribution.
+$$H_0:\mu=\mu_0$$
+$$H_1:\mu>\mu_0$$
+$$\mu_0=4$$
+$$n=16$$
+$$\bar Y=4.2$$
+$$S=0.8$$
+$$\alpha=0.1$$
+$$t^*=\sqrt n(\bar Y-\mu_0)/S=$$
+$$t_{(n-1,\alpha)}=t_{(15,0.1)}=$$
+
+\bigskip
+11b.
+$$H_0:\sigma^2=\sigma_0^2$$
+$$H_1:\sigma^2<\sigma_0^2$$
+$$\sigma_0^2=0.64$$
+$${\chi^2}^*=(n-1)S^2/\sigma_0^2=$$
+
+\vfill
+\eject
+
+12. Variance unknown but equal, normal distribution.
+$$H_0:\mu_1-\mu_2=0$$
+$$H_1:\mu_1-\mu_2\ne0$$
+$$\bar X=$$
+$$S_X=$$
+$$\bar Y=$$
+$$S_Y=$$
+$$\alpha=0.05$$
+
+\end