Chapter 10.1, Problem 24E

(a)

To determine

To explain does this study convincing evidence of a difference at the $\alpha =0.05$ significance level of a difference in effectiveness of Lipitor and Pravachol for people like the ones in this study.

Answer to Problem 24E

There is convincing evidence that there is a difference in the effectiveness of Lipitor and Pravachol for people like the ones in this study.

Explanation of Solution

Given:

  $\begin{array}{l}{n}_1=2063\\ n_2=2099\\ {\widehat{p}}_1=0.263\\ {\widehat{p}}_2=0.224\\ \alpha =0.05\end{array}$

Define Hypotheses:

So, the given claim that: difference between proportions.

Now, we have to find out the appropriate hypotheses for performing a significance test.

Thus, the claim is either the null hypothesis or the alternative hypothesis. The null hypothesis states that the population proportions are equal. If the null hypothesis is the claim then the alternative hypothesis states the opposite of the null hypothesis.

Therefore, the appropriate hypotheses for this is:

  $\begin{array}{l}{H}_0:p_1=p_2\\ H_a:p_1\ne p_2\end{array}$

Where we have,

  $p_1=$ the proportion of people who take Pravacholand died within two years.

  $p_2=$ the proportion of people who take Lipitor and died within two years.

Conditions to be satisfied:

There are three conditions to be satisfied:

Random: It is satisfied because the volunteers are independent random samples assigned to the treatment groups.

Independent: It is satisfied because the $2063$ people with heart disease are less than $10\%$ of all people with heart disease and $2099$ people with heart disease are less than $10\%$ of all people with heart disease.

Normal: It is satisfied because

  $\begin{array}{l}{n}_1{\widehat{p}}_1=2063(0.263)\\ =542.569\\ n_1(1-{\widehat{p}}_1)=2063(1-0.263)\\ =2063(0.737)\\ =1520.431\\ n_2{\widehat{p}}_2=2099(0.224)\\ =470.176\\ n_2(1-{\widehat{p}}_2)=2099(1-0.224)\\ =2099(0.776)\\ =1628.824\end{array}$

are all at least ten.

Thus, all the conditions are satisfied.

Calculation:

The sample proportion is the number of successes divided by the sample size. Then, we have,

  $\begin{array}{l}{\widehat{p}}_1=\frac{x_1}{n_1}\\ \Rightarrow x_1=n_1{\widehat{p}}_1\\ =2063(0.263)\\ =542.569\\ \approx 543\\ {\widehat{p}}_2=\frac{x_2}{n_2}\\ \Rightarrow x_2=n_2{\widehat{p}}_2\\ =2099(0.224)\\ =470.176\\ \approx 470\\ {\widehat{p}}_p=\frac{x_1+x_2}{n_1+n_2}\\ =\frac{543+470}{2063+2099}\\ =\frac{1013}{4162}\\ =0.2434\end{array}$

Now, we will calculate the value of test statistics as:

  $\begin{array}{l}z=\frac{{\widehat{p}}_1-{\widehat{p}}_2-(p_1-p_2)}{\sqrt{{\widehat{p}}_p(1-{\widehat{p}}_p)}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\\ =\frac{0.263-0.224-0}{\sqrt{0.2434(1-0.2434)}\sqrt{\frac{1}{2063}+\frac{1}{2099}}}\\ \approx 2.93\end{array}$

The P-value is the probability of obtaining the value of the test statistics or a value more extreme assuming that the null hypothesis is true. Thus, we have,

  $\begin{array}{l}P=P(Z<-2.93\text{ or }Z>2.93)\\ =2P(Z<-2.93)\\ =2(0.0017)\\ =0.0034\end{array}$

Thus, if the P-value is smaller than the significance level, then we will reject the null hypothesis, thus, we have,

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

Thus, we conclude that there is convincing evidence that there is a difference in the effectiveness of Lipitor and Pravachol for people like the ones in this study.

(b)

To determine

To explain based on your conclusion in part (a), which mistake −Type I error or Type II error you could have made.

Answer to Problem 24E

Type I error.

Explanation of Solution

In part (a), the conclusion was made that there is convincing evidence that there is a difference in the effectiveness of Lipitor and Pravachol for people like the ones in this study. And the hypothesis defined was that:

  $\begin{array}{l}{H}_0:p_1=p_2\\ H_a:p_1\ne p_2\end{array}$

Thus, based on the conclusion in part (a), we have:

Type I error is the error when we reject the null hypothesis, when the null hypothesis is false.

And the Type II error is when we fail to reject the null hypothesis, when the null hypothesis is false.

Since in this case we rejected the null hypothesis, then it can mean that we have made a Type I error.

(c)

To determine

To explain should you generalize your results in part (a) to all people with heart disease.

Answer to Problem 24E

Thus, it cannot be generalize to all people with heart disease.

Explanation of Solution

As the results in part (a),

We can say that as the study used volunteers and it is possible that these volunteers have common characteristics.

For example, it is possible that the people with the most severe heart disease volunteer because they do not have any effective medicine available for them. This then implies that the death rates for the volunteers could be much higher than for all people with heart disease in general. So, it is then not appropriate to generalize the results of the volunteers in the study to all people with heart disease.