Chapter 10, Problem R10.3RE

(a)

To determine

To find out do the data provide convincing evidence at the $\alpha =0.05$ level that taking AZT lowers the proportion of infected people like the ones in this study who will develop AIDS ina given period of time or not.

Answer to Problem R10.3RE

Yes.

Explanation of Solution

It is given that:

  $\begin{array}{l}{x}_1=38\\ x_2=17\\ n_1=n_2=435\end{array}$

Let us now define the hypotheses:

  $\begin{array}{l}{H}_0:p_1=p_2\\ H_{\alpha }:p_1>p_2\end{array}$

The sample proportion is the number of successes divided by the sample size:

  $\begin{array}{l}{\widehat{p}}_1=\frac{x_1}{n_1}\\ =\frac{38}{435}\\ =0.0874\\ {\widehat{p}}_2=\frac{x_2}{n_2}\\ =\frac{17}{435}\\ =0.0391\\ {\widehat{p}}_p=\frac{x_1+x_2}{n_1+n_2}\\ =\frac{38+17}{435+435}\\ =\frac{55}{870}\\ =0.0632\end{array}$

Thus, the value of test statistics is:

  $\begin{array}{l}z=\frac{{\widehat{p}}_1-{\widehat{p}}_2}{\sqrt{{\widehat{p}}_p(1-{\widehat{p}}_p)}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\\ =\frac{0.0874-0.0391}{\sqrt{0.0632(1-0.0632)}\sqrt{\frac{1}{435}+\frac{1}{435}}}\\ =2.93\end{array}$

The P -value is the probability of obtaining the value of the test statistic or a value more extreme. Then, determine the P -value using table A:

  $\begin{array}{l}P=P(Z>2.93)\\ =P(Z<-2.93)\\ =0.0017\end{array}$

If the P -value is smaller than the significance level, reject the null hypothesis, thus we have,

  $P<0.05\Rightarrow \text{ Reject }{H}_0$

Thus, we can say that there is sufficient evidence to support the claim. Therefore, the data provide convincing evidence at the $\alpha =0.05$ level that taking AZT lowers the proportion of infected people like the ones in this study who will develop AIDS in a given period of time.

(b)

To determine

To describe the Type I and Type II error in this setting and give a consequence of each error.

Explanation of Solution

Let us now define the hypotheses:

  $\begin{array}{l}{H}_0:p_1=p_2\\ H_{\alpha }:p_1>p_2\end{array}$

Now, we know that the Type I error is:

Reject the null hypothesis $H_0$ , when $H_0$ is true.

It interprets that the drug seems to be effective while it is not effective.

The consequence of this error is that people will take the drug without getting better.

Now, for Type II error, this error occurs when:

Fail to reject the null hypothesis $H_0$ , when $H_0$ is false.

It interprets that the drug seems to be not effective while it is effective.

The consequence of this error will be that the drug will not come on the market because the drug seemed to be not effective.

Type I error is more serious because people will take a drug without getting better.