Question

Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question.

Measured Results vs. Reported Results The same study cited in the preceding exercise produced these results after six months for the 198 patients given sustained care: 25.8% were no longer smoking, and these results were biochemically confirmed, but 40.9% of these patients

reported that they were no longer smoking. Construct the two 95% confidence intervals. Compare the results. What do you conclude?

Short answer

95% confidence interval for non-smokers is between 19.7% and 31.9%, while the proportion of non-smokers biomedically confirmed non-smokers has 95% confidence interval is between 34.0% and 47.7%.

The two confidence intervals do not overlap with each other.

From the confidence interval, it can be stated that sustained care is more effective than standard care.

Step-by-step solution

Given information

The given claim is about the program that helped the patients to stop smoking. There were 198(n) patients who were given sustained care, 25.8% $\left({\hat{p}}_1\right)$ did not smoke and it was biochemically confirmed. And 40.9% $\left({\hat{p}}_2\right)$of these patients reported that they no longer smoke.

Requirements for computing confidence interval

The basic requirement of this confidence interval of proportion is that the sample is a simple random sample.

This requirement is assumed to be satisfied.

Assuming fixed trials and constant rate of success.

It is required to check,$\mathbf{np}⩾\mathbf{5}\mathbf{and}\mathbf{nq}⩾\mathbf{5}$ for each case.

$$\begin{gathered} n{\hat{p}}_1=198\times 0.258 \\ =51.084 \\ ⩾5 \end{gathered}$$

$$\begin{gathered} n{\hat{q}}_1=198\times \left(1-0.258\right) \\ =146.92 \\ ⩾5 \end{gathered}$$

Also,

$$\begin{gathered} n{\hat{p}}_2=198\times 0.409 \\ =80.982 \\ ⩾5 \end{gathered}$$

$$\begin{gathered} n{\hat{q}}_2=198\times \left(1-0.409\right) \\ =117.02 \\ ⩾5 \end{gathered}$$

Thus, the requirements for the test are satisfied.

State the formula for confidence interval 

The Confidence Interval for proportion is calculated as below,

$\hat{p}-E<\hat{p}<\hat{p}+E$;

The margin of error can be computed by the formula given below,

$E=z_{\frac{\alpha }{2}}\times \sqrt{\frac{\hat{p}\hat{q}}{n}}$

For 95% confidence interval, the critical value is obtained from standard normal table as 1.96$\left(z_{\frac{0.05}{2}}\right)$ .

Compute the confidence interval for sustained care 

Margin of error for sustained care no longer smoking are,

$$\begin{gathered} E=z_{\frac{\alpha }{2}}\times \sqrt{\frac{{\hat{p}}_1{\hat{q}}_1}{n_1}} \\ =1.96\times \sqrt{\frac{0.258\times 0.742}{198}} \\ =0.061 \end{gathered}$$

Substitute all the values in the formula.

$$\begin{gathered} \text{Confidence}\text{interval}={\hat{p}}_1-E<\hat{p}<{\hat{p}}_1+E \\ =0.258-0.061<\hat{p}<0.258+0.061 \\ =0.197<\hat{p}<0.319 \end{gathered}$$

The 95% confidence interval for proportion is between 19.7% and 31.9%.

Compute the confidence interval for biochemically tested results

Margin of error for standard care,

$$\begin{gathered} E=z_{\frac{\alpha }{2}}\times \sqrt{\frac{{\hat{p}}_2{\hat{q}}_2}{n_2}} \\ =1.96\times \sqrt{\frac{0.409\times 0.591}{198}} \\ =0.0685 \end{gathered}$$

Substitute all the values in the formula.

$$\begin{gathered} \text{Confidence}\text{interval}={\hat{p}}_2-E<\hat{p}<{\hat{p}}_2+E \\ =0.409-0.068<\hat{p}<0.409+0.068 \\ =0.340<\hat{p}<0.477 \end{gathered}$$

The 95% confidence interval for proportion is between 34.0% and 47.7%.

Analyze the intervals 

The values range differently, as they are non-overlapping.

It can be concluded that the proportion of patients confirmed no longer smoking biomedically was higher.