Assignment 3

1. Coin Toss Hypothesis Test

• Given: $n=900$, $\alpha =0.05$, $p=0.5$
• Mean: $\mu =np=900\times 0.5=450$
• Std Dev: $\sigma =\sqrt{np(1-p)}=\sqrt{900\times 0.5\times 0.5}=15$
• z-critical: $z_{\alpha /2}=1.96$
• Acceptance region:
  • $450\pm 1.96(15)=[420.6,479.4]$
• Conclusion: Reject $H_0$ if heads $\le 420$ or $\ge 480$

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2. Guinea Pig Birth-Weight Test

• Data: $n=27$, $\bar{x}=325.4963$ g, $s=198.7855$ g

(a) Hypothesis Test ($\alpha =0.05$)

• Null: $H_0:\mu =300$
• Test stat:
  • $t=\frac{325.4963-300}{198.7855/\sqrt{27}}\approx 0.6665$
• df: $n-1=26$, $t_{0.025,26}\approx 2.0555$
• Conclusion: $|t|<2.0555$ → Fail to reject $H_0$

(b) Smallest $\alpha$

• p-value: $p=2(1-F_{t_{26}}(0.6665))\approx 0.5110$
• Conclusion: Only reject $H_0$ if $\alpha >0.511$

(c) 95% Confidence Interval

• $\bar{x}\pm t_{0.025,26}\cdot \frac{s}{\sqrt{n}}$
• $325.4963\pm 2.0555\cdot \frac{198.7855}{\sqrt{27}}$
• CI $\approx (246.86,404.13)$ g

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3. Proportion of Rough Bearings

• Given: $n=85$, $x=10$ → $\hat{p}=10/85\approx 0.1176$

(a) Test $H_0:p=0.10$ vs $H_1:p>0.10$

• $z=\frac{\hat{p}-0.10}{\sqrt{0.10\cdot 0.90/85}}\approx 1.715$
• $z_{0.05}=1.645$ → $1.715>1.645$ ⇒ Reject $H_0$
• $p$-value $\approx 0.0433$

(b) Power calculation at $p=0.15$

• Fail region: $\hat{p}\le 0.11694$
• $Z^{\prime }=\frac{0.11694-0.15}{\sqrt{0.15\cdot 0.85/85}}\approx -0.8537$
• $\beta =\Phi (-0.8537)\approx 0.196$

(c) Sample size for 90% power at $p=0.15$

• $n\approx 362$

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4. Comparing Two Formulations

• Given: ${\sigma }_1^2=1.5$, ${\sigma }_2^2=1.2$, $n_1=15$, ${\bar{x}}_1=89.6$, $n_2=20$, ${\bar{x}}_2=92.5$

(a) Hypothesis Test $H_0:{\mu }_2\le {\mu }_1$, $H_1:{\mu }_2>{\mu }_1$

• $z=\frac{92.5-89.6}{\sqrt{1.5/15+1.2/20}}=7.25$
• $z_{0.05}=1.645$ → Reject $H_0$
• $p$ value $\approx 3\times 10^{-13}$

(b) 95% CI on ${\mu }_2-{\mu }_1$

• CI: $2.9\pm 1.645(0.4)=(2.242,3.558)$

(c) Sample size for $\text{error}<1$ at 95% CL

• $n\approx 11$ per group

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5. Two-sample $t$-test: Catalyst Yield

• Catalyst 1: $n_1=12$, ${\bar{y}}_1=86$, $s_1=3$
• Catalyst 2: $n_2=15$, ${\bar{y}}_2=89$, $s_2=2$

(a) Test at $\alpha =0.01$

• $s_p^2=\frac{11\cdot 9+14\cdot 4}{25}=6.2$, $s_p=2.49$
• SE = $s_p\cdot \sqrt{1/n_1+1/n_2}\approx 0.964$
• $t=\frac{89-86}{0.964}\approx 3.112$
• $t_{0.01,25}=2.492$ → Reject $H_0$

(b) 99% CI on ${\mu }_2-{\mu }_1$

• CI = $3\pm 2.787\cdot 0.964=(0.313,5.687)$

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6. Variance Comparison (F-test)

• Men: $n_m=25$, $s_m=0.98$; Women: $n_w=21$, $s_w=1.02$

(a) Hypothesis Test $H_0:{\sigma }_m^2={\sigma }_w^2$, $H_1:{\sigma }_m^2\ne {\sigma }_w^2$

• $F=\frac{0.98^2}{1.02^2}=0.923$
• $d{f}_1=24$, $d{f}_2=20$
• Critical values: $F_{0.99,24,20}\approx 2.54$, lower $\approx 0.39$
• $0.39<0.923<2.54$ → Fail to reject $H_0$

(b) 98% CI for $\frac{{\sigma }_m^2}{{\sigma }_w^2}$

• CI: $\left(\frac{0.923}{2.54},0.923\times 2.54\right)\approx (0.36,2.42)$