10.8 Review Questions

  1. In a poll of 1961 randomly selected U.S. adults, 1137 said that they do not believe that abstinence programs are effective in reducing or preventing AIDS.
    1. At the 2% significance level, do the data provide sufficient evidence to conclude that a majority of all U.S. adults feel that way?
    2. Obtain a 98% confidence interval for the percentage of U.S. adults who believe that abstinence programs are effective in reducing or preventing AIDS.
    3. Interpret the interval obtained in part (b). Does it support the result of the test in part (a)?
  2. In a clinical trial, 56 patients were randomly assigned to use the Bug Buster kit and 70 were assigned to use the standard treatment. Thirty-two patients in the Bug Buster kit group were cured, whereas nine of those in the standard treatment group were cured.
    1. At the 5% significance level, do these data provide sufficient evidence to conclude that a difference exists in the cure rates of the two types of treatment?
    2. Determine a 95% confidence interval for the difference in cure rates for the two types of treatment.
    3. Interpret the interval obtained in part (b). Does it support the result of the test in part (a)?
Show/Hide Answer
  1. ‌
    1. Check the assumptions:
      • we have a simple random sample
      • Sample size assumption: [latex]np_0=1961\times0.5=980.5>5, n(1-p_0)=1961\times (1-0.5)=980.5>5[/latex].

      Steps: we have [latex]n=1961, x=1137, \hat p=\frac{x}{n}=\frac{1137}{1961}=0.5798.[/latex]

      1. Hypotheses: [latex]H_0: p\le 0.5[/latex] versus [latex]H_a: p>0.5[/latex].
      2. Significance level [latex]\alpha=0.02[/latex].
      3. Test statistics.
        [latex]z_o=\frac{\hat p-p_0}{\sqrt{p_0\times (1-p_0)/n}}=\frac{\frac{1137}{1961}-0.5}{\sqrt{0.5\times (1-0.5)/1961}}=7.068.[/latex]
      4. [latex]P\mbox{-value}[/latex]: For a right-tailed test, [latex]P\mbox{-value}[/latex]=[latex]P(Z\ge 7.068)=P(Z\le -7.068)\approx 0.[/latex]
      5. Decision: We reject [latex]H_0[/latex] since [latex]P\mbox{-value}[/latex][latex]\approx 0<0.02 (\alpha)[/latex].
      6. Conclusion: At the 2% significance level, we have sufficient evidence that a majority of all U.S. adults do not believe that abstinence programs are effective in reducing or preventing AIDS.
    2. [latex]n=1961, x=1137, \hat p=\frac{x}{n}=\frac{1137}{1961}=0.5798, 1-\alpha=0.98,[/latex] [latex]\alpha=0.02, z_{\alpha/2}=z_{0.01}=2.33.[/latex]
      A 98% confidence interval for the proportion of U.S. adults who do not believe that abstinence programs are effective in reducing or preventing AIDS is
      [latex]\hat p \pm z_{\frac{\alpha}{2}}\sqrt{\frac{\hat p(1-\hat p)}{n}}=0.5798 \pm 2.33 \times \sqrt{\frac{0.5798 (1-0.5798)}{1961}}=(0.5538, 0.6057)[/latex]
      which gives the percentage is somewhere between 55.38% and 60.57%.
    3. Interpretation: we can be 98% confident that the percentage of U.S. adults who do not believe that abstinence programs are effective in reducing or preventing AIDS is somewhere between 55.38% and 60.57%.
      Yes, we reject [latex]H_0[/latex] and claim that [latex]p>0.5[/latex] in the hypothesis test. The entire 98% confidence interval is above 0.5, so we can claim that [latex]p>0.5[/latex].
  2. ‌
    1. Check the assumptions:
      • we have simple random samples from the two treatment groups.
      • the two samples are independent.
      • [latex]x_1=32, n_1-x_1=56-32=24[/latex], [latex]x_2=9[/latex], and [latex]n_2-x_2=70-9=61[/latex] are all greater than 5.

      Steps:

      1. Hypotheses: [latex]H_0: p_1-p_2= 0[/latex] versus [latex]H_a: p_1-p_2\ne 0[/latex].
      2. Significance level [latex]\alpha=0.05[/latex].
      3. Test statistics.
        [latex]z_o=\frac{(\hat p_1-\hat p_2)-0}{\sqrt{\hat p_{\scriptsize p}(1-\hat p_{\scriptsize p})}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}=\frac{0.5714-0.1286}{\sqrt{0.3254(1-0.3254)}\sqrt{\frac{1}{56}+\frac{1}{70}}}=5.271[/latex]
        where the pooled proportion is given by
        [latex]\hat p_{\scriptsize p}=\frac{x_1+x_2}{n_1+n_2}=\frac{32+9}{56+70}=0.3254[/latex] ; [latex]\hat p_1=\frac{x_1}{n_1}=\frac{32}{56}=0.5714[/latex] ; [latex]\hat p_2=\frac{x_2}{n_2}=\frac{9}{70}=0.1286[/latex].
      4. [latex]P\mbox{-value}[/latex]: For a two-tailed test, [latex]P\mbox{-value}[/latex]=[latex]2P(Z\ge 5.271)=2P(Z\le -5.271)\approx 0.[/latex]
      5. Decision: We reject [latex]H_0[/latex] since [latex]P\mbox{-value}[/latex][latex]\approx 0<0.05 (\alpha)[/latex].
      6. Conclusion: At the 5% significance level, we have sufficient evidence that a difference exists in the cure rates of the two types of treatment.
    2. ‌
      [latex]\begin{aligned} & (\hat p_1-\hat p_2)\pm z_{\frac{\alpha}{2}}\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2}}\\ &=(0.5714-0.1286)\pm 1.96\times \sqrt{\frac{0.5714(1-0.5714)}{56}+\frac{0.1286(1-0.1286)}{70}} \\ &=(0.2913, 0.5943). \end{aligned}[/latex]
    3. Interpretation: We can be 95% confident that the cure rate of the Bug Buster kits group is 0.2913 to 0.5943 higher than the cure rate of the standard treatment.
      Yes, since the entire interval is above 0, we can claim that [latex]p_1-p_2>0[/latex], which is the conclusion of the hypothesis in part (a).

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