Question

Excedrin or Tylenol? In a study to compare the effects of two pain relievers it was found that of \(n_{1}=200\) randomly selected individuals who used the first pain reliever, \(93 \%\) indicated that it relieved their pain. Of \(n_{2}=450\) randomly selected individuals who used the second pain reliever, \(96 \%\) indicated that it relieved their pain. a. Find a \(99 \%\) confidence interval for the difference in the proportions experiencing relief from pain for these two pain relievers. b. Based on the confidence interval in part a, is there sufficient evidence to indicate a difference in the proportions experiencing relief for the two pain relievers? Explain.

Short answer

Answer: No, there is insufficient evidence to indicate a difference in the proportions experiencing relief for the two pain relievers with 99% confidence, as the confidence interval (-0.0153, 0.0753) contains 0.

Step-by-step solution

Calculate the proportions of pain relief for both relievers

First, find the proportion for each pain reliever by dividing the number of individuals who experienced pain relief by the total number of individuals using the pain reliever. For pain reliever 1: P1 = 0.93 For pain reliever 2: P2 = 0.96

Calculate the difference in proportions

Next, find the difference in proportions between the two pain relievers: Difference = P2 - P1 = 0.96 - 0.93 = 0.03

Calculate the standard error of the difference in proportions

Now, we'll find the standard error of the difference in proportions: Standard Error = sqrt[(P1 * (1 - P1) / n1) + (P2 * (1 - P2) / n2)] Plugging in the values, we get: Standard Error = sqrt[(0.93 * (1 - 0.93) / 200) + (0.96 * (1 - 0.96) / 450)] ≈ 0.0176

Find the 99% confidence interval for the difference in proportions

For a 99% confidence interval, we will use a Z-score of 2.576 (obtained from the Z-table or a calculator). Confidence Interval = (Difference - Z-score * Standard Error, Difference + Z-score * Standard Error) Confidence Interval = (0.03 - 2.576 * 0.0176, 0.03 + 2.576 * 0.0176) Confidence Interval ≈ (-0.0153, 0.0753)

Analyze the confidence interval to determine if there is sufficient evidence to indicate a difference

The 99% confidence interval for the difference in proportions is (-0.0153, 0.0753). Because the interval contains 0, we cannot conclude that there is a significant difference in the proportions experiencing relief for the two pain relievers with 99% confidence. So, there is insufficient evidence to indicate a difference in the proportions.

Key concepts

Statistical Inference

Statistical inference is the process of using data from a sample to make generalizations about a larger population. In the context of the given exercise, statistical inference allows us to estimate the difference in effectiveness between two pain relievers based on the experiences of a sample of individuals.

To do this, we assume that the sample is representative of the population, meaning the outcomes we observed in the sample are likely to occur in the broader population. However, we acknowledge there is uncertainty in these estimates, which is where confidence intervals and hypothesis testing come into play.

Confidence Interval Calculation

The confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence. For example, a 99% confidence interval for the difference in proportions means we are 99% confident that the true difference between the population proportions falls within this interval.

To calculate this interval, we first determine the difference in sample proportions. Then, we use the standard error of the difference to adjust for variability. Finally, a Z-score associated with our desired confidence level (99% in this case) is used to widen the interval to an appropriate range that acknowledges the level of uncertainty in our estimate.

Proportion Comparison

When comparing proportions, such as the percentage of individuals who experienced pain relief from two different pain relievers, we're interested in whether a significant difference exists between them. To assess this, we not only compute the point estimate of the difference but also account for the variability in each proportion.

In comparing the proportions from our exercise, we calculated the 99% confidence interval around the estimated difference. The intention is to evaluate whether the observed difference could be due to chance or if it reflects a true difference in efficacy between the two pain relievers.

Hypothesis Testing

Hypothesis testing is a systematic method for evaluating statistical evidence in a sample. In the context of proportion comparison, the null hypothesis typically states that there is no difference between the population proportions. The alternative hypothesis suggests that a difference does exist.

In our exercise, analyzing the confidence interval informs the hypothesis test. Because zero, which indicates no difference, is within our confidence interval, we fail to reject the null hypothesis. There is not enough statistical evidence to conclude that a difference exists between the pain relief effectiveness of the two drugs at the 99% confidence level.