Chapter PVI, Problem 14RE

(a)

To determine

To create a $95\%$ confidence interval for the difference in proportions of smokers in the two groups.

Answer to Problem 14RE

We are $95\%$ confident that the difference in proportions of smokers in the two groups lie between $(0.0394,0.1266)$ .

Explanation of Solution

It is given in the question that in the depression and heart attack research described in previous exercise, $32\%$ of the diseased group were smokers, compared with only $23.7\%$ of those free of heart disease. Thus, to find the confidence interval for proportion we will use the calculator $\text{TI}-89$ . To calculate a confidence interval for a population proportion: Go to the STAT Intsmenu and select $6:2-$ PropZInt. Enter the observed counts and the sample size for both the samples. Specify a confidence level. Calculate the interval. So, we have,

  $\begin{array}{l}{p}_1=0.32\\ n_1=450\\ p_2=0.237\\ n_2=2397\end{array}$

And $\alpha =0.05$ so by using the calculator $\text{TI}-89$ , the confidence interval be as:

  $(0.0394,0.1266)$ .

So, we are $95\%$ confident that the difference in proportions of smokers in the two groups lie between $(0.0394,0.1266)$ .

(b)

To determine

To explain is this evidence that the two groups in the study were different.

Answer to Problem 14RE

The two groups in the study were different.

Explanation of Solution

It is given in the question that in the depression and heart attack research described in previous exercise, $32\%$ of the diseased group were smokers, compared with only $23.7\%$ of those free of heart disease.Thus, to do the mechanics of a hypothesis test for equality of population proportions we will use the calculator $\text{TI}-89$ . Then select $6:2-$ PropZTestfrom the STAT Tests menu. Enter the observed counts and sample sizes. Indicate what kind of test you want: one-tail upper tail, lower tail, or two-tail. And specify whether results should simply be calculated or displayed with the area corresponding to the P-value of the test shaded. So, we have,

  $\begin{array}{l}{p}_1=0.32\\ n_1=450\\ p_2=0.237\\ n_2=2397\end{array}$

And the hypothesis is defined as:

  $\begin{array}{l}{H}_0:p_{\text{1}}-p_{\text{2}}=0\\ H_a:p_{\text{1}}-p_{\text{2}}\ne 0\end{array}$

Thus, by using the calculator $\text{TI}-89$ , the test statistics and the P-value is as:

  $\begin{array}{l}z=3.73\\ P=0.0002\end{array}$

As we know that if the P-value is less than or equal to the significance level then the null hypothesis is rejected, so we have,

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

Thus, we conclude thatthere is sufficient evidence that the two groups in the study were different.

(c)

To determine

To explain could this be a problem in analysing the results of the study.

Answer to Problem 14RE

Yes, this could be a problem in analysing the results of the study.

Explanation of Solution

It is given in the question that in the depression and heart attack research described in previous exercise, $32\%$ of the diseased group were smokers, compared with only $23.7\%$ of those free of heart disease. Yes, there is problem in analysing the results of the study because according to the conclusion in part (b), we know that they were different so, how can we will be able to compare the two groups and conclude to a conclusion.