Question

In an experiment to treat patients with "generalized anxiety disorder," the drug hydroxyzine was given to 71 patients, and 30 of them improved. A group of 70 patients were given a placebo, and 20 of them improved. \(^{51}\) Let \(p_{1}\) and \(p_{2}\) represent the probabilities of improvement using hydroxyzine and the placebo, respectively. Construct a \(95 \%\) confidence interval for \(\left(p_{1}-p_{2}\right)\).

Short answer

The 95% confidence interval for \(p_1 - p_2\) is \(\hat{p}_1 - \hat{p}_2 \pm 1.96 \cdot SE\), where \(SE = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}\).

Step-by-step solution

Calculate the sample proportions

First, we calculate the sample proportions for both groups. For the hydroxyzine group, the sample proportion of improvement is \(p_1 = \frac{\text{number of improvements with hydroxyzine}}{\text{total number of patients given hydroxyzine}} = \frac{30}{71}\). Similarly, for the placebo group, \(p_2 = \frac{\text{number of improvements with placebo}}{\text{total number of patients given placebo}} = \frac{20}{70}\).

Calculate the standard error (SE)

The standard error of the difference in proportions is calculated by using the formula: \(SE = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}\), where \(n_1\) and \(n_2\) are the sample sizes for both groups.

Find the z-value for the 95% confidence level

To construct a 95% confidence interval, we need to find the z-value corresponding to the middle 95% of the data. Found from a standard normal distribution table or calculator, the z-value that leaves 2.5% in each tail is approximately 1.96.

Construct the confidence interval

The confidence interval for \(p_1 - p_2\) can be constructed with the formula \(\hat{p}_1 - \hat{p}_2 \pm z\cdot SE\). Use the z-value obtained from Step 3 and the standard error calculated in Step 2 to find the confidence interval.

Calculate and interpret the interval

Finally, calculate the confidence interval by substituting the values into the formula. The resulting interval provides the range within which the true difference in treatment effect between hydroxyzine and placebo lies with 95% confidence.

Key concepts

Generalized Anxiety Disorder Treatment Study

When researching treatments for disorders such as generalized anxiety disorder (GAD), scientists often conduct controlled studies to compare the effectiveness of different interventions. In this context, a common approach is to compare a therapeutic drug with a placebo. A well-designed study on GAD treatment may involve randomly assigning patients to receive either the experimental drug or the placebo, then tracking which patients exhibit signs of improvement. The goal is to determine the efficacy of the drug in treating symptoms relative to the placebo, providing insights into whether the observed effects are due to the medication or not. It's critical to have a large enough sample size and to follow proper protocols to reduce bias and increase the validity of the study's findings.

Sample Proportions Calculation

When working with experimental data, such as in the GAD treatment study, researchers calculate sample proportions to quantify outcomes. The sample proportion is the ratio of individuals in a sample who exhibit a particular trait—improvement from GAD symptoms in this case—divided by the total number of individuals in that sample. To calculate the sample proportion, use the formula:
\[\begin{equation}p = \frac{\text{number of individuals with the trait}}{\text{total number of individuals in the sample}}\end{equation}\]
For both the drug hydroxyzine and the placebo, these proportions help to determine how effective each treatment was within the sample studied. The resulting values are crucial for further statistical analysis, like the confidence interval we want to construct.

Standard Error of Difference in Proportions

In statistics, to compare the difference between two sample proportions, such as the effectiveness of two treatments, we compute the standard error of the difference in proportions. The standard error provides a measure of the variability or standard deviation of the sampling distribution of the difference between two proportions. This is calculated with the formula:
\[\begin{equation}SE = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}\end{equation}\]
where \( p_1 \) and \( p_2 \) are the sample proportions of each group and \( n_1 \) and \( n_2 \) are their respective sample sizes. A smaller SE suggests a more precise estimate of the true difference in population proportions.

Z-value for Confidence Level

In constructing confidence intervals, the 'z-value' plays a critical role. It corresponds to the desired confidence level—in our case, 95%. The z-value tells us how many standard errors to extend from the point estimate to capture the central percentage of the normal distribution. It is sourced from the standard normal distribution (Z-distribution).

For a 95% confidence interval, the z-value approximately equals 1.96, which means that the interval extends 1.96 standard errors above and below the sample proportion difference. This z-value is selected to leave a 2.5% tail on each end of the distribution, encompassing the central 95% of the distribution.

Statistical Inference

Statistical inference involves using data from a sample to make conclusions about a larger population. The confidence interval is a core tool in this process. It's used to estimate the range within which a population parameter, based on sample statistics, is likely to lie. For instance, a 95% confidence interval for the difference in proportions between two treatments indicates that if we were to take many samples and build intervals in the same way, we'd expect about 95% of those intervals to contain the true difference in population proportions.

In the context of the GAD treatment study, we infer from our constructed interval the likely difference in the effectiveness of hydroxyzine versus a placebo. It's a powerful method that encapsulates both the estimate of the difference and the uncertainty associated with it.