Question

Treating AIDS The drug AZT was the first drug that seemed effective in delaying the onset of AIDS. Evidence for AZT’s effectiveness came from a large randomized comparative experiment. The subjects were $870$volunteers who were infected with HIV, the virus that causes AIDS, but did not yet have AIDS. The study assigned $435$of the subjects at random to take $500$milligrams of AZT each day and another $435$to take a placebo. At the end of the study, $38$of the placebo subjects and $17$of the AZT subjects had developed AIDS.

(a) Researchers want to test the claim that taking AZT lowers the proportion of infected people who will develop AIDS in a given period of time. Carry out a test of this claim at the $\alpha =0.05$ level.

(b) The experiment was double-blind. Explain why this is important. (c) Describe a Type I and a Type II error in this setting, and give the consequences of each. Which is more serious?

Short answer

a). Yes.

b). A double-blind is one of the experiments in which neither the individuals nor the subjects who measure know which treatment was received.

c). Type I- error is the most serious one.

Step-by-step solution

Part (a) Step 1: Given Information

$x_1=38$

$x_2=17$

$n_1=n_2=435$

Part (a) Step 2: Explanation

To find: out the test of the claim at the $\alpha =0.05$level

localid="1650369535354" $H_0:p_1=p_2$

$H_1:p_1>p_2$

The sample proportion is

${\hat{p}}_1=\frac{x_1}{n_1}$

$=\frac{38}{435}=0.0874$

${\hat{p}}_2=\frac{x_2}{n_2}$

localid="1650390409222" $$\begin{gathered} =\frac{17}{435} \\ =0.0391 \end{gathered}$$

${\hat{p}}_p=\frac{x_1+x_2}{n_1+n_2}$

$=\frac{38+17}{435+435}$

$=0.0632$

Part (a)  Step 3: Explanation

The test-statistics is

$z=\frac{{\hat{p}}_1+{\hat{p}}_2}{\sqrt{{\hat{p}}_p\left(1-{\hat{p}}_p\right)}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$

$=\frac{0.0874-0.0391}{\sqrt{0.0632(1-0.0632)}\sqrt{\frac{1}{435}+\frac{1}{435}}}$

$=2.93$

P-value is the probability of getting the value of test statistic. The P-value is

$P=P(Z>2.93)$

$=P(Z<-2.93)$

localid="1650357089811" $=0.0017$

When the P-value is smaller than the significance level, then reject the null hypothesis. $P<0.05$therefore, here is enough proof to help the claim.

Part (b) Step 1: Given Information

At the end of the study, $38$ of the placebo subjects and $17$ of the AZT subjects had developed AIDS.

Part (b) Step 2: Explanation

A double-blind is one of the experiments in which neither the individuals nor the subjects who measure know which treatment was received, where in the single-blind experiment either the individuals who measure or the subjects know which treatment was received. Therefore, it is important for the subjects, such that it is chance to eliminate the placebo effect. This is also good for the individuals who execute who execute the measurements.

Part (c) Step 1: Given Information

$H_0:p_1=p_2$

localid="1650357109301" $H_1:p_1>p_2$

Part (c) Step 2: Explanation

Type I error: reject the null hypothesis if the null hypothesis

The drug appears to be effective, whereas it is not effective. People would take the drug without getting better Type Il error: fail to reject the null hypothesis, if the null hypothesis is false.

The drug appears to be not effective if it is effective. The drug would not come on the market, the reason is that the drug appears to be not effective.

Therefore Type-l error is the most serious one because people would take a drug without getting better.