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Chapter 21: Problem 34

The data you used in the previous two problems came from a random sample of students who took the SAT twice. The response rate was \(63 \%\), which is pretty good for nongovernment surveys, so let's accept that the respondents do represent all students who took the exam twice. Nonetheless, we can't be sure that coaching actually caused the coached students to gain more than the uncoached students. Explain briefly but clearly why this is so.

Short Answer

Expert verified
The lack of random assignment and the presence of confounding variables prevent proving causation in this observational study.

Step by step solution

01

Understand Sampling vs. Causation

Even though the sample is random and the response rate is high, it doesn't prove causation. A high response rate can mean that the data is representative, but without random assignment to treatments, we can't establish a causal relationship.
02

Identify the Role of Confounding Variables

Other variables, known as confounders, might influence the outcomes observed. For example, students who chose coaching might differ systematically from those who didn't, perhaps in motivation or prior knowledge, affecting the test outcome independently of the coaching.
03

Recognize Observational Study Limitations

This scenario is based on an observational study, not an experiment. In observational studies, we observe outcomes without assigning treatments randomly, which limits our ability to infer cause-and-effect relationships, such as whether coaching directly causes improved SAT scores.

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Key Concepts

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

Causation vs Correlation
Students often grapple with the difference between causation and correlation, especially in observational studies. Simply put, correlation indicates that two variables move in relation to each other, but it doesn't mean one causes the other. For instance, if coached students seem to score higher on the SAT, it doesn't automatically mean coaching caused the better scores; this might just be a correlation.
  • Causation implies a direct relationship where one event is the result of the occurrence of the other event.
  • Correlation, however, simply shows that two variables are related, but without implying cause.

To establish causation, one needs to control for all other variables and ideally conduct a randomized controlled trial. Observational studies, like the one mentioned, typically don't allow for such strict control, hence no direct causation can be claimed.
Random Sampling
A random sample is crucial when you want to draw conclusions about a larger population from a smaller subset. It involves selecting individuals at random, reducing biases and making the data more representative. In the case of the SAT study, students were randomly selected to ensure the sample reflected the diversity of all students taking the test twice.
  • This helps in generalizing findings, ensuring they're not skewed by non-random factors.
  • However, while random sampling aids in representativity, it doesn't solve the causation question on its own.

Even with random sampling, without random assignment to control or experimental groups, one cannot disentangle the possible influences of other factors besides the one being studied.
Confounding Variables
Confounding variables are the hidden influences in an observational study that could affect the results. They are variables that the researcher failed to control or eliminate, damaging the study's inner validity. For example, in the context of the SAT study, students might self-select into coaching because they are more motivated or have more resources. These attributes themselves might lead to higher scores.
  • Confounders provide alternative explanations for observed results.
  • They create a link between the independent and dependent variables that the study might not account for.
  • This leads to issues in determining whether the observed changes in the dependent variable are caused by the independent variable or the confounders.

Identifying and accounting for confounding variables remains a major challenge in observational research, and failing to do so might lead to erroneous conclusions about causation.

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Most popular questions from this chapter

What is the effect of concussions on the brain? Researchers measured the brain sizes (hippocampal volume in microliters) of 25 collegiate football players with a history of clinician-diagnosed concussion and 25 collegiate football players without a history of concussion. Here are the summary statistics: 18 $$ \begin{array}{lccc} \hline \text { Group } & \text { Group Size } & \text { Mean } & \text { Standard Deviation } \\ \hline \text { Concussion } & 25 & 5784 & 609.3 \\ \hline \text { Nonconcussion } & 25 & 6489 & 815.4 \\ \hline \end{array} $$ (a) Is there evidence of a difference in mean brain size between football players with a history of concussion and those without concussions? (b) The researchers in this study stated that participants were "consecutive cases of healthy National Collegiate Athletic Association Football Bowl Subdivision Division I football athletes with \((n=25)\) or without \((n=25)\) a history of clinician-diagnosed concussion ... between June 2011 and August 2013" at a U.S. psychiatric research institute specializing in neuroimaging among collegiate football players. What effect does this information have on your conclusions in part (a)?

In the 2015 NAEP sample of 12th-graders in the United States, the mean mathematics scores were 150 for female students and 153 for male students. To see if this difference is statistically significant, you would use (a) the two-sample \(t\) test. (b) the matched pairs \(t\) test. (c) the one-sample \(t\) test.

Does the way the press depicts the economic future affect the stock market? To investigate this, researchers analyzed the longest article about the crisis from the front page of the Money section of USA Today from a randomly chosen weekday of each week between August 2007 and June 2009. Articles were rated as to how positive or negative they were about the economic future. For each week, the change in the Dow Jones Industrial Average (average DJIA value the week after the article appeared minus the average the week the article appeared) was computed. Positive values of the change indicate that the DJIA increased. \({ }^{21}\) Here are the changes in DJIA corresponding to very positive articles: \(\begin{array}{rrrrrrr}-325 & -200 & -225 & -75 & -25 & 25 & 50 \\ 225 & 25 & -225 & -250 & 200 & 250 & 75\end{array}\) Here are the changes in DJIA values corresponding to very negative articles: $$ \begin{array}{rrrrrrr} 150 & 300 & 225 & 125 & -175 & -225 & -375 \\ -175 & 0 & 125 & 175 & 475 & & \end{array} $$ Is there good evidence that the DJIA performs differently after very positive articles than after very negative articles? (a) Do the sample means suggest that there is a difference in the change in the DJIA after very positive articles versus after very negative articles? (b) Make stemplots for both samples. Are there any obvious departures from Normality? (c) Test the hypothesis \(H_{0}: \mu_{1}=\mu_{2}\) against the two-sided alternative. What do you conclude from part (a) and from the result of your test? (d) Among a host of different factors that are claimed to have triggered the economic crisis of 2007-2009, one was a "culture of irresponsibility" in the way the future was depicted in the press. Do the data provide any evidence that negative articles in the press contributed to poor performance of the DJIA?

Do education programs for preschool children that follow the Montessori method perform better than other programs? A study compared five-year-old children in Milwaukee, Wisconsin, who had been enrolled in preschool programs from the age of three. \({ }^{15}\) (a) Explain why comparing children whose parents chose a Montessori school with children of other parents would not show whether Montessori schools perform better than other programs. (In fact, all the children in the study applied to the Montessori school. The school district assigned students to Montessori or other preschools by a random lottery.) (b) In all, 54 children were assigned to the Montessori school and 112 to other schools at age three. When the children were five, parents of 30 of the Montessori children and 25 of the others could be located and agreed to and subsequently participated in testing. This information reveals a possible source of bias in the comparison of outcomes. Explain why. (c) One of the many response variables was score on a test of ability to apply basic mathematics to solve problems. Here are summaries for the children who took this test: $$ \begin{array}{lccc} \hline \text { Group } & n & \mathrm{x}^{-\bar{x}} & \mathrm{~s} \\ \hline \text { Montessori } & 30 & 19 & 3.11 \\ \hline \text { Control } & 25 & 17 & 4.19 \\ \hline \end{array} $$ Is there evidence of a difference in the population mean scores? (The researchers used two-sided alternative hypotheses.)

The 2015 National Assessment of Educational Progress (NAEP) gave a mathematics test to a random sample of 12th-graders in the United States. The mean score was 152 out of 300 . To give a confidence interval for the mean score of all 12th-graders in the United States, you would use (a) the two-sample \(t\) interval. (b) the matched pairs \(t\) interval. (c) the one-sample \(t\) interval.
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