Question

In a study to compare the effects of two pain relievers it was found that of \(n_{1}=200\) randomly selectd individuals instructed to use the first pain reliever, \(93 \%\) indicated that it relieved their pain. Of \(n_{2}=450\) randomly selected individuals instructed to use 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 not sufficient evidence to indicate a difference in the proportions experiencing relief from the two pain relievers, as the 99% confidence interval (-0.1141, 0.0541) includes the value 0, indicating no difference.

Step-by-step solution

Calculate proportions

For the first pain reliever, p1 = (93% of 200) / 200 = 0.93. For the second pain reliever, p2 = (96% of 450) / 450 = 0.96.

Calculate the difference in proportions

We need to find the difference between the two proportions: D = p1 - p2 = 0.93 - 0.96 = -0.03.

Find the z-score for 99% confidence

For a 99% confidence interval, we will use the standard z-score of 2.576 (rounded to 3 decimals).

Compute the standard error

The standard error for the sampling difference of two proportions is given by the formula: SE = sqrt((p1*(1-p1)/n1) + (p2*(1-p2)/n2)) = sqrt((0.93*(1-0.93)/200) + (0.96*(1-0.96)/450)) ≈ 0.0326

Calculate the confidence interval

Using the z-score and standard error, we can determine the confidence interval as follows: CI = D ± z*SE = -0.03 ± 2.576*0.0326 ≈ (-0.1141, 0.0541)

Analyze the confidence interval

The 99% confidence interval calculated for the difference in the proportions experiencing relief from pain for these two pain relievers is (-0.1141, 0.0541). Since 0, the value indicating no difference, is within the confidence interval, we cannot conclude that there is sufficient evidence to show that the two pain relievers have different effects on the proportions experiencing relief.

Key concepts

Confidence Interval

A confidence interval provides a range of values that we expect the true difference in proportions to lie within, rather than a single estimate. In this example, we are interested in the difference in relief provided by two pain relievers.
A 99% confidence interval means we are 99% certain that the true difference lies within our computed range. The process involves multiple steps:

• First, we find the sample proportions: 0.93 for the first and 0.96 for the second pain reliever.
• Next, calculate the standard error, which accounts for the variability of the sample proportions.
• Then, determine the z-score for a 99% confidence interval, which is 2.576.
• Finally, apply the z-score to the standard error to find the confidence interval, which is approximately (-0.1141, 0.0541).

Since the interval includes 0, there's no strong evidence to suggest the two pain relievers are different significantly in terms of pain relief.

Proportions

Proportions in statistics measure how much of a whole is represented by a part. In our exercise, it’s the number of people relieved by each pain reliever divided by the total number of people who tried it.
We have:

• For the first pain reliever: 93% of 200 people were relieved, giving us a proportion of 0.93.
• For the second pain reliever: 96% of 450 people were relieved, resulting in a proportion of 0.96.

Understanding these proportions is crucial because it lays the groundwork for determining if there’s any difference in effectiveness. A greater proportion generally suggests better effectiveness, but we also need to consider other statistical factors, as seen in calculating confidence intervals.

Hypothesis Testing

Hypothesis testing is a method of deciding whether the evidence from a sample is strong enough to infer a statement about a population. In this case, we want to determine if there is a truly significant difference between the pain relief rates of the two medications.
Steps in our hypothesis testing include:

• Formulate the null hypothesis (no difference in effectiveness) and the alternative hypothesis (there is a difference).
• Calculate the difference in sample proportions, which is -0.03 in our study.
• Determine if this difference is statistically significant by evaluating the computed confidence interval.
• If the confidence interval includes 0, we retain the null hypothesis.

Since our interval includes 0, we conclude there isn't enough evidence to prove a difference between the effectiveness of the two pain relievers.

Z-score

A z-score is a statistical measurement that quantifies the deviation of a data point from the mean, measured in standard deviations. In the context of confidence intervals, it helps determine how confidently we can assert the range in which a parameter lies.
For a 99% confidence interval, the z-score is 2.576. Here's how it works:

• This z-score is multiplied by the standard error to provide the margin of error.
• The margin of error determines how wide or narrow our confidence interval will be.
• A higher z-score leads to a wider interval, reflecting more certainty in our range but less precision.

The choice of a 99% level reflects a desire for high confidence, making it less likely that our interval misses the true difference, though this results in a broader range.