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.

OxyContinThe drug OxyContin (oxycodone) is used to treat pain, but it is dangerous because it is addictive and can be lethal. In clinical trials, 227 subjects were treated with OxyContin and 52 of them developed nausea (based on data from Purdue Pharma L.P.).

a.Construct a 95% confidence interval estimate of the percentageof OxyContin users who develop nausea.

b.Compare the result from part (a) to this 95% confidence interval for 5 subjects who developed nausea among the 45 subjects given a placebo instead of OxyContin: 1.93% <p< 20.3%. What do you conclude?

Short answer

a. The confidence interval is from 17.44% to 28.37%.

b. OxyContin and Placebo have different rates of effect on developing nausea.

Step-by-step solution

Given information

The numbers of subjects treated with OxyCotin were recorded. The number sample size, $n=227$. The confidence interval is 95%

Check the requirement

a.

1. The samples are selected randomly.

2. There are two categories of outcomes, either developed nausea or not.

3. The count of success is 52 and failures is 175 which areboth are greater than 5.

Therefore, the conditions are satisfied.

Calculate the sample proportion

Here we need to estimate the confidence interval of the OxyCotin users who developed nausea.

Thesample proportion of theOxyContin userswho developed nausea is:

$$\begin{gathered} \hat{p}=\frac{x}{n} \\ =\frac{52}{227} \\ =0.2291 \end{gathered}$$

Therefore, the sample proportion is 0.2291.

Then,

$$\begin{gathered} \hat{q}=1-\hat{p} \\ =1-0.2291 \\ =0.7709 \end{gathered}$$

Compute the critical value

At $95\%$confidence interval $\alpha =0.05$,

Using the standard normal table, the critical value is computed as,

$$\begin{gathered} z_{crit}=z_{\frac{\alpha }{2}} \\ =z_{0.025} \\ =1.96 \end{gathered}$$

Compute margin of error

The margin of error is given by,

$$\begin{gathered} E=z_{crit}\times \sqrt{\frac{\hat{p}\hat{q}}{n}} \\ =1.96\times \sqrt{\frac{0.23\times 0.77}{227}} \\ =0.0547 \end{gathered}$$

Compute the confidence interval

The formula for the confidence interval is given by,

$$\begin{gathered} CI=\hat{p}-E<p<\hat{p}+E \\ =\left(0.23-0.0547<p<0.23+0.0547\right) \\ =(0.1744<p<0.2837) \end{gathered}$$

In percentage, the 95% confidence interval is $17.44\%<p<28.37\%$

Conclusion

b.

The confidence interval in the case of the treatment group is between 17.44% to 28.37%. The 95% confidence interval in the case of the control group is between 1.93% and 20.3%.

Each of the intervals are different from each other and they do not overlap. Hence it is concluded that OxyContin and Placebo have different rates of effect on developing nausea.