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Researchers comparing the effectiveness of two pain medications randomly selected a group of patients who had been complaining of a certain kind of joint pain. They randomly divided these people into two groups, then administered the pain killers. Of the 112 people in the group who received medication A, 84 said this pain reliever was effective. Of the 108 people in the other group, 66 reported that pain reliever B was effective. a. Write a \(95 \%\) confidence interval for the percent of people who may get relief from this kind of joint pain by using medication A. Interpret your interval. b. Write a \(95 \%\) confidence interval for the percent of people who may get relief by using medication B. Interpret your interval. c. Do the intervals for \(A\) and B overlap? What do you think this means about the comparative effectiveness of these medications? d. Find a \(95 \%\) confidence interval for the difference in the proportions of people who may find these medications effective. Interpret your interval. e. Does this interval contain zero? What does that mean? f. Why do the results in parts \(c\) and e seem contradictory? If we want to compare the effectiveness of these two pain relievers, which is the correct approach? Why?

Step by step solution

01

Calculate Confidence Interval for Medication A

First, calculate the proportion of people who reported that Medication A was effective, this is \( p_A = 84/112 = 0.75 \). Then, calculate the standard error of proportion using the formula \(SE = \sqrt{ p_A (1 - p_A) / n_A }\) where \( p_A\) is the proportion of success, \( n_A = 112\) is the total number of participants. The 95% confidence interval is given by \( p_A \pm 1.96*SE \). Plug in the values to get the confidence interval.

02

Calculate Confidence Interval for Medication B

Calculate the proportion of people who reported that Medication B was effective, this is \( p_B = 66 / 108 = 0.611 \). Then, calculate the standard error as above but use \( p_B\) and \(n_B = 108\). The 95% confidence interval for B is given by \( p_B \pm 1.96*SE \). Plug in the numbers to get the confidence interval.

03

Compare the Confidence Intervals

Check if the confidence intervals for A and B overlap. If they do, this suggests that there is no significant difference between the effectiveness of the two medications.

04

Calculate confidence interval for the difference in effect

Calculate the difference in proportions \( d = p_A - p_B \) and the standard error of the difference \( SE = \sqrt{SE_A^2 + SE_B^2}\). The 95% confidence interval for the difference in proportions is given by \( d \pm 1.96*SE \). Plug in the values to get the confidence interval.

05

Check if the interval contains zero

If the confidence interval for the difference includes zero, it suggests that there is no significant difference in the effectiveness of the two medications.

06

Resolve the apparent contradiction

The apparent contradiction between steps 3 and 5 may be due to different assumptions underlying the two methods. The appropriate method to use when comparing the effectiveness of two treatments depends on the research question and the nature of the data. Here, the difference in proportions is more appropriate as it directly measures the difference in effectiveness.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proportion Comparison

When comparing two different treatments or groups, it's essential to evaluate the proportion of success or effectiveness in each. In our exercise with pain medications A and B, we determine how many participants found each medication effective. To find the proportion, we take the number of successful outcomes (patients who found relief) and divide it by the total number of participants in each group. For example, if 84 out of 112 people said medication A worked for them, the proportion is \[ p_A = \frac{84}{112} = 0.75. \] Similarly, for medication B, if 66 out of 108 people found it effective, the proportion is \[ p_B = \frac{66}{108} = 0.611. \] Understanding these proportions is the first step in comparing the effectiveness of two different treatments.

Standard Error

The standard error gives us an idea of how much variability we can expect in our sample proportions due to random sampling error. It helps us assess the reliability of our sample estimates.To calculate the standard error for a proportion, we use the formula: \[ SE = \sqrt{\frac{p(1-p)}{n}}, \] where \( p \) is the sample proportion, and \( n \) is the sample size. The smaller the standard error, the more reliable the estimate of the proportion.For medication A, the calculation is:\[ SE_A = \sqrt{\frac{0.75 \times (1 - 0.75)}{112}}. \]For medication B, it is:\[ SE_B = \sqrt{\frac{0.611 \times (1 - 0.611)}{108}}. \]These standard errors are crucial for constructing confidence intervals, which provide a range of possible values for the true proportions in the population.

95% Confidence Level

A confidence interval gives us a range of values within which we expect the true population parameter to lie, with a certain degree of confidence. Here, we're focusing on a 95% confidence level, meaning we are 95% sure that the true proportion of people who find relief from the medication falls within our interval.To calculate a 95% confidence interval, we take the sample proportion \( p \), and use the formula: \[ p \pm 1.96 \times SE, \] where \( 1.96 \) is the z-score corresponding to a 95% confidence level. Applying it to medication A:The confidence interval is: \[ 0.75 \pm 1.96 \times SE_A. \]And for medication B:\[ 0.611 \pm 1.96 \times SE_B. \]These intervals help us understand the effectiveness range for each medication based on our sample data.

Statistical Significance

Statistical significance helps us determine whether the observed difference between two groups is likely due to chance or reflects a genuine difference in the population. When comparing two proportions, like those of medications A and B, we are interested in whether the difference is statistically significant.Firstly, we calculate the difference between the two proportions: \[ d = p_A - p_B. \]Then, we compute the standard error for this difference: \[ SE_d = \sqrt{SE_A^2 + SE_B^2}. \]Next, we create a confidence interval for the difference:\[ d \pm 1.96 \times SE_d. \]If this interval includes zero, it indicates that there is no statistically significant difference in the effectiveness of the two medications. This means that any observed difference in proportions may just be due to random sampling error. Understanding significance is key in determining whether one medication can be confidently considered more effective than the other.