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Chapter 10: Problem 74

Exercises 71 to 74 refer to the following setting. Coaching companies claim that their courses can raise the SAT scores of high school students. Of course, students who retake the SAT without paying for coaching generally raise their scores. A random sample of students who took the SAT twice found 427 who were coached and 2733 who were uncoached.\(^{44}\) Starting with their Verbal scores on the first and second tries, we have these summary statistics: Coaching and SAT scores: Critique (4.1, 4.3) The data you used in the previous two exercises came from a random sample of students who took the SAT twice. The response rate was 63%, which is pretty good for nongovernment surveys. (a) Explain how nonresponse could lead to bias in this study. (b) We can鈥檛 be sure that coaching actually caused the coached students to gain more than the un- coached students. Explain briefly but clearly why this is so.

Short Answer

Expert verified
Nonresponse bias and confounding variables could skew results, making conclusions about coaching's effectiveness uncertain.

Step by step solution

01

Understand Nonresponse Bias

Nonresponse bias occurs when individuals who do not respond to a survey differ in important ways from those who do participate. In this study, the response rate is 63%, which means 37% did not respond. If the students who did not respond have systematically different SAT score improvements (perhaps because they lacked motivation or failed to improve even with coaching), this could skew the results, making the coaching appear more or less effective than it truly is.
02

Consider Confounding Variables

In studies like these, it is crucial to consider confounding variables鈥攆actors other than the one being studied that could affect the outcome. For example, motivated students are more likely to opt for coaching and also more likely to study harder on their own, resulting in greater score improvements. Thus, any observed improvement might not be caused solely by the coaching.
03

Analyze the Causation Claim

Even if coached students show greater improvements, this does not necessarily prove that coaching caused the improvement. Other explanations, such as students spending more time preparing in general or having better initial verbal skills, could account for the differences in score gains. Without a controlled experimental design, causation cannot be established conclusively.

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Key Concepts

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

Nonresponse Bias in SAT Score Studies
In any study, it's important to get responses from as many participants as possible to ensure accurate results. However, in this SAT score study, there was a 37% nonresponse rate, which hints at nonresponse bias. Nonresponse bias happens when people who don鈥檛 respond to a survey differ significantly from those who do respond.
For example, students who didn鈥檛 respond might have different reasons for not improving their SAT scores, such as lack of motivation or ineffective coaching. When these students are not represented in the data, the results could misrepresent the effectiveness of coaching.
This bias may lead to incorrect conclusions about the coaching鈥檚 impact. For instance, if non-responding students generally didn鈥檛 improve much, the study might overestimate the success rate of the coaching programs among responding students. Therefore, understanding and mitigating nonresponse bias is crucial in interpreting the study results accurately.
Role of Confounding Variables
Confounding variables are those sneaky factors that can muddy the waters of research conclusions. In the SAT score improvement study, several external factors beyond coaching may influence a student's score enhancement.
Consider motivated students who choose coaching programs. Their drive to improve might encourage them to engage in self-study more earnestly than their peers. Thus, their potentially higher test scores might be attributed to personal study habits rather than the coaching alone.
Additionally, students with ample resources (like access to more study materials or private tutors) might naturally perform better. Therefore, when analyzing the coaching effect, it's pivotal to identify and control for these confounding variables to avoid misleading results.
Understanding Causation in Studies
Grasping the difference between correlation and causation is fundamental in evaluating research findings. In cases like this SAT score improvement study, even if data shows coached students scored better than uncoached peers, it doesn鈥檛 prove coaching is the reason for the improvement.
Without a controlled experimental setup isolating coaching as the sole variable, it's difficult to exclude other contributing factors such as prior preparedness, additional study practices, or intrinsic student abilities.
To firmly claim causation, one would need a study where participants are randomly assigned to either receive coaching or not, thus minimizing the impact of other factors. Otherwise, researchers can only suggest that coaching correlates with improvement, leaving the actual cause uncertain.

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Most popular questions from this chapter

Sleep deprivation Does sleep deprivation linger for more than a day? Researchers designed a study using 21 volunteer subjects between the ages of 18 and 25. All 21 participants took a computer-based visual discrimination test at the start of the study. Then the subjects were randomly assigned into two groups. The 11 subjects in one group, D, were deprived of sleep for an entire night in a laboratory setting. The 10 subjects in the other group, A, were allowed unrestricted sleep for the night. Both groups were allowed as much sleep as they wanted for the next two nights. On Day 4, all the subjects took the same visual discrimination test on the computer. Researchers recorded the improvement in time (measured in milliseconds) from Day 1 to Day 4 on the test for each subject.\(^{41}\) We used Fathom software to randomly reassign the 21 subjects to the two groups 1000 times, assuming the treatment received doesn鈥檛 affect each individual鈥檚 time improvement on the test. The dotplot shows the approximate randomization distribution of \(\overline{x}_{\mathrm{A}}-\overline{x}_{\mathrm{D}}\). (a) Explain why the researchers didn鈥檛 let the subjects choose whether to be in the sleep deprivation group or the unrestricted sleep group. (b) In the actual experiment, \(\overline{x}_{\mathrm{A}}-\overline{x}_{\mathrm{D}}=15.92 .\) This value is marked with a blue line in the figure. What conclusion would you draw? Justify your answer with appropriate evidence. (c) Based on your conclusion in part (b), could you have made a Type I error or a Type II error? Justify your answer.

Multiple choice: Select the best answer for Exercises 29 to 32. A sample survey interviews SRSs of 500 female college students and 550 male college students. Each student is asked whether he or she worked for pay last summer. In all, 410 of the women and 484 of the men say 鈥淵es.鈥� Exercises 29 to 31 are based on this survey. The pooled sample proportion who worked last summer is about (a) \(\hat{p}_{\mathrm{C}}=1.70 . \quad(\mathrm{d}) \hat{p}_{\mathrm{C}}=0.85\) (b) \(\hat{p}_{\mathrm{C}}=0.89 . \quad\) (e) \(\hat{p}_{\mathrm{C}}=0.82\) (c) \(\hat{p}_{\mathrm{C}}=0.88\)

Young adults living at home A surprising number of young adults (ages 19 to 25) still live in their parents鈥� homes. A random sample by the National Institutes of Health included 2253 men and 2629 women in this age group.\(^{11}\) The survey found that 986 of the men and 923 of the women lived with their parents. (a) Construct and interpret a 99% confidence interval for the difference in population proportions (men minus women). (b) Does your interval from part (a) give convincing evidence of a difference between the population proportions? Explain.

Paired or unpaired? In each of the following settings, decide whether you should use paired \(t\) procedures or two-sample t procedures to perform inference. Explain your choice.\(^{42}\) (a) To test the wear characteristics of two tire brands, A and B, each brand of tire is randomly assigned to 50 cars of the same make and model. (b) To test the effect of background music on productivity, factory workers are observed. For one month, each subject works without music. For another month, the subject works while listening to music on an MP3 player. The month in which each subject listens to music is determined by a coin toss. (c) A study was designed to compare the effectiveness of two weight-reducing diets. Fifty obese women who volunteered to participate were randomly assigned into two equal-sized groups. One group used Diet A and the other used Diet B. The weight of each woman was measured before the assigned diet and again after 10 weeks on the diet.

Exercises 71 to 74 refer to the following setting. Coaching companies claim that their courses can raise the SAT scores of high school students. Of course, students who retake the SAT without paying for coaching generally raise their scores. A random sample of students who took the SAT twice found 427 who were coached and 2733 who were uncoached.\(^{44}\) Starting with their Verbal scores on the first and second tries, we have these summary statistics: Coaching and SAT scores (10.1) What proportion of students who take the SAT twice are coached? To answer this question, Jannie decides to construct a 99% confidence interval. Her work is shown below. Explain what鈥檚 wrong with Jannie鈥檚 method. $$hat{p}_{1}=\frac{427}{3160}=0.135=\underset{\text { who were coached }}{\text { proportion of students }}$$ $$\hat{p}_{2}=\frac{2733}{3160}=0.865=\underset{\text { who weren't coached }}{\text { who weren't coached }}$$ $$\mathrm{A} 99 \% \mathrm{Cl} \text { for } p_{1}-p_{2} \mathrm{is}$$ $$\begin{aligned}(0.135-0.865) \pm 2.575 \sqrt{\frac{0.135(0.865)}{3160}} &+\frac{0.865(0.135)}{2733} \\ &=-0.73 \pm 0.022=(-0.752,-0.708) \end{aligned}$$ We are 99% confident that the proportion of students taking the SAT twice who are coached is between 71 and 75 percentage points lower than students who aren鈥檛 coached.
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