/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 43 Researchers comparing the effect... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 20: Problem 43

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?

Short Answer

Expert verified
Medication A and B do not significantly differ in effectiveness according to the comparison of confidence intervals as they overlap. However, the 95% confidence interval for the difference does not contain zero, suggesting a significant difference in effectiveness. This apparent contradiction can be resolved by noting that the latter method is more appropriate for this comparison as it directly measures the difference in effectiveness.

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A new vaccine was recently tested to see if it could prevent the painful and recurrent ear infections that many infants suffer from. The Lancet, a medical journal, reported a study in which babies about a year old were randomly divided into two groups. One group received vaccinations; the other did not. During the following year, only 333 of 2455 vaccinated children had ear infections, compared to 499 of 2452 unvaccinated children in the control group. a. Are the conditions for inference satisfied? b. Find a \(95 \%\) confidence interval for the difference in rates of ear infection. c. Use your confidence interval to explain whether you think the vaccine is effective.

In Chapter 6 , Exercise 25 , we looked at collected samples of water from streams in the Adirondack Mountains to investigate the effects of acid rain. Researchers measured the pH (acidity) of the water and classified the streams with respect to the kind of substrate (type of rock over which they flow). A lower pH means the water is more acidic. Here is a boxplot of the \(\mathrm{pH}\) of the streams by substrate (limestone, mixed, or shale): Here are selected parts of a software analysis comparing the pH of streams with limestone and shale substrates: 2 -Sample \(t\) -Test of \(\mu_{1}-\mu_{2}\) Difference Between Means \(=0.735\) \(t\) -Statistic \(=16.30 \mathrm{w} / 133 \mathrm{df}\) \(\mathrm{p} \leq 0.0001\) a. State the null and alternative hypotheses for this test. b. From the information you have, do the assumptions and conditions appear to be met? c. What conclusion would you draw?

A Vermont study published by the American Academy of Pediatrics examined parental influence on teenagers' decisions to smoke. A group of students who had never smoked were questioned about their parents' attitudes toward smoking. These students were questioned again two years later to see if they had started smoking. The researchers found that, among the 284 students who indicated that their parents disapproved of kids smoking, 54 had become established smokers. Among the 41 students who initially said their parents were lenient about smoking, 11 became smokers. Do these data provide strong evidence that parental attitude influences teenagers' decisions about smoking? a. What kind of design did the researchers use? b. Write appropriate hypotheses. c. Are the assumptions and conditions necessary for inference satisfied? d. Test the hypothesis and state your conclusion. e. Explain in this context what your P-value means. \(\mathrm{f}\). If it is later found that parental attitudes actually do influence teens' decisions to smoke, which type of

A presidential candidate fears he has a problem with women voters. His campaign staff plans to run a poll to assess the situation. They'll randomly sample 300 men and 300 women, asking if they have a favorable impression of the candidate. Obviously, the staff can't know this, but suppose the candidate has a positive image with \(59 \%\) of males but with only \(53 \%\) of females. a. What kind of sampling design is his staff planning to use? b. What difference would you expect the poll to show? c. Of course, sampling error means the poll won't reflect the difference perfectly. What's the standard deviation for the difference in the proportions? d. Sketch a sampling model for the size difference in proportions of men and women with favorable impressions of this candidate that might appear in a poll like this. e. Could the campaign be misled by the poll, concluding that there really is no gender gap? Explain.

The Journal of the American Medical Association reported on an experiment intended to see if the drug Prozac \(^{\circledast}\) could be used as a treatment for the eating disorder anorexia nervosa. The subjects, women being treated for anorexia, were randomly divided into two groups. Of the 49 who received Prozac, 35 were deemed healthy a year later, compared to 32 of the 44 who got the placebo. a. Are the conditions for inference satisfied? b. Find a \(95 \%\) confidence interval for the difference in outcomes. c. Use your confidence interval to explain whether you think Prozac is effective.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.