Question

Lowering bad cholesterol Refer to Exercise 24.

a. Construct and interpret a $95\%$ confidence interval for the difference between the true proportions. Assume that the conditions for inference are met.

b. Explain how the confidence interval provides more information than the test in

Exercise 24.

Short answer

a. Confidence Interval is $\left(0.0129,0.0651\right)$

b. It is because Confidence Interval gives us range of possible values.

Step-by-step solution

Given Information

It is given that $n_1=2063$

$n_2=2099$

${\hat{p}}_1=0.263$

${\hat{p}}_2=0.224$

$\alpha =0.05$

Confidence Interval

All three conditions of random, independent and normality are met.

For $1-\alpha =0.95$, from table $z_{u/2}=1.96$

Confidence interval: $\left({\hat{p}}_1-{\hat{p}}_2\right)-z_{a/2}\times \sqrt{\frac{{\hat{p}}_1\left(1-{\hat{p}}_1\right)}{n_1}+\frac{{\hat{p}}_2\left(1-{\hat{p}}_2\right)}{n_2}}$

$=(0.263-0.224)-1.96\times \sqrt{\frac{0.263(1-0.263)}{2063}+\frac{0.224(1-0.224)}{2099}}\approx 0.0129$

and $\left({\hat{p}}_1-{\hat{p}}_2\right)+z_{\alpha /2}\times \sqrt{\frac{{\hat{p}}_1\left(1-{\hat{p}}_1\right)}{n_1}+\frac{{\hat{p}}_2\left(1-{\hat{p}}_2\right)}{n_2}}\approx 0.0651$

Hence, we are $95\%$confident that true death rate of people taking Prachachol lies between$0.0129\text{and}0.0651$higher than true death rate of people who take Lipitor.

How Confidence Interval is better

Confidence interval is better as it gives us range of possible values. Significance test only checks one difference value of difference between proportions,