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Solutions Manual for ‘Computational Probability and Statistics’

4 Studies

4.1 Objectives

  1. Define and use properly in context all new terminology, to include: confounding variable, prospective study, retrospective study, simple random sampling, stratified sampling, strata, cluster sampling, multistage sampling, experiment, randomized experiment, control, replicate, blocking, blocks, treatment group, control group, blinded study, placebo, placebo effect, and double-blind.

  2. Given a study description, be able to describe the study using correct terminology.

  3. Given a scenario, describe flaws in reasoning and propose study and sampling designs.

4.2 Homework

4.2.1 Problem 1

Propose a sampling strategy. A large college class has 160 students. All 160 students attend the lectures together, but the students are divided into 4 groups, each with 40 students, for lab sections administered by different teaching assistants. The professor wants to conduct a survey about how satisfied the students are with the course, and he believes that the lab section a student is in might affect the student’s overall satisfaction with the course.

  1. What type of study is this?

Observational study. However, this could easily be implemented as an experiment, if students were randomized to teaching assistant/lab section as a sort of treatment.

  1. Suggest a sampling strategy for carrying out this study.

A stratified random sample would work well, where we sample randomly within each lab section.

4.2.2 Problem 2

Flawed reasoning. Identify the flaw in reasoning in the following scenarios. Explain what the individuals in the study should have done differently if they want to be able to make such strong conclusions.

  1. Students at an elementary school are given a questionnaire that they are required to return after their parents have completed it. One of the questions asked is, Do you find that your work schedule makes it difficult for you to spend time with your kids after school? Of the parents who replied, 85% said no. Based on these results, the school officials conclude that a great majority of the parents have no difficulty spending time with their kids after school.

Non-responders may have a different response to this question. The parents who returned the surveys are probably those who do not have difficulty spending time with their kids after school. Parents who work might not have returned the surveys since they probably have a busier schedule. There might also be some response bias in those that did respond, leading parents to try to “impress” the school and answer no even if they experience difficulty spending time with their kids after school.

  1. A survey is conducted on a simple random sample of 1,000 women who recently gave birth, asking them about whether or not they smoked during pregnancy. A follow-up survey asking if the children have respiratory problems is conducted 3 years later, however, only 567 of these women are reached at the same address. The researcher reports that these 567 women are representative of all mothers.

It is unlikely that the women who were reached at the same address 3 years later are a representative sample. These missing respondents are probably renters (as opposed to homeowners), which means that they might be in a lower socio-economic status than the respondents who were reached at the same address. Additionally, there may be some response bias in the respondents because women may be reluctant to admit they smoked during pregnancy.

4.2.3 Problem 3

Sampling strategies. A statistics student who is curious about the relationship between the amount of time students spend on social networking sites and their performance at school decides to conduct a survey. Four research strategies for collecting data are described below. In each, name the sampling method proposed and any bias you might expect. Note: Sampling methods from both Chapter 3 (Overview of Data Collection Principles) and Chapter 4 (Studies) may be used for this problem.

  1. He randomly samples 40 students from the study’s population, gives them the survey, and asks them to fill it out and bring it back the next day.

This is a simple random sample. Non-response bias is a concern; if only those people who have strong opinions about the survey respond, his sample may not be representative of the population.

  1. He gives out the survey only to his friends, and makes sure each one of them fills out the survey.

This is a convenience sample. Undercoverage bias is a concern; his sample may not be representative of the population since it consists only of his friends. It is also possible that the study will have non-response bias if some choose to not bring back the survey.

  1. He posts a link to an online survey on his Facebook wall and asks his friends to fill out the survey.

This is a convenience sample, and a voluntary sample. This will have a similar issues to handing out surveys to friends. The survey is likely to have undercoverage bias and non-response bias. There is also likely to be bias stemming from the fact that this is a voluntary sample; his friends have to volunteer to take the online survey, and it is often the most opinionated people who choose to participate in online (or phone, etc.) surveys.

  1. He stands outside the QRC and asks every third person that walks out the door to fill out the survey.

This is a systematic sample. This could have non-response bias if students leaving the QRC decide not to participate in the survey. It could also have undercoverage bias, as every third student may not be enough separation between students to get good coverage.

4.2.4 Problem 4

Vitamin supplements. In order to assess the effectiveness of taking large doses of vitamin C in reducing the duration of the common cold, researchers recruited 400 healthy volunteers from staff and students at a university. A quarter of the patients were assigned a placebo, and the rest were evenly divided between 1g Vitamin C, 3g Vitamin C, or 3g Vitamin C plus additives to be taken at the onset of a cold for the following two days. All tablets had identical appearance and packaging. The nurses who handed the prescribed pills to the patients knew which patient received which treatment, but the researchers assessing the patients when they were sick did not. No significant differences were observed in any measure of cold duration or severity between the four medication groups, and the placebo group had the shortest duration of symptoms.

  1. Is this an experiment or an observational study? Why?

This is an experiment, because the researchers randomly assigned different treatments to the participants.

  1. What are the explanatory and response variables in this study?

The response variable is the duration of the cold.
The explanatory variable is the treatment, with 4 levels; placebo, 1g, 3g, and 3g with additives.

  1. Were the patients blinded to their treatment?

The patients were blinded as they did not know which treatment they received.

  1. Was this study double-blind?

The study was double-blind with respect to the researchers evaluating the patients, but the nurses who briely interacted with patients during the distribution of the medication were not blinded. (It was partially double-blind.)

  1. Participants are ultimately able to choose whether or not to use the pills prescribed to them. We might expect that not all of them will adhere and take their pills. Does this introduce a confounding variable to the study? Explain your reasoning.

Since the patients were randomly assigned to the treatment groups and they are blinded, we would expect about an equal number of patients in each group to not adhere to the treatment. While this means that final results of the study will be based on fewer number of participants, non-adherence does not introduce a confounding variable to the study.

4.2.5 Problem 5

Exercise and mental health. A researcher is interested in the effects of exercise on mental health and she proposes the following study: Use stratified random sampling to ensure representative proportions of 18-30, 31-40 and 41-55 year olds from the population. Next, randomly assign half the subjects from each age group to exercise twice a week, and instruct the rest not to exercise. Conduct a mental health exam at the beginning and at the end of the study, and compare the results.

  1. What type of study is this?

This is an experiment since we assigned subjects to the exercise program.

  1. What are the treatment and control groups in this study?

The treatment is exercise twice a week and the control is no exercise.

  1. Does this study make use of blocking? If so, what is the blocking variable?

Yes, the blocking variable is age.

  1. Does this study make use of blinding?

No, the study is not blinded since the patients will know whether or not they are exercising.

  1. Comment on whether or not the results of the study can be used to establish a causal relationship between exercise and mental health, and indicate whether or not the conclusions can be generalized to the population at large.

Since this is an experiment, we can make a causal statement. Since the sample is random, the causal statement can be generalized to the population at large. However, we should be cautious about making a causal statement because of a possible placebo effect.

  1. Suppose you are given the task of determining if this proposed study should get funding. Would you have any reservations about the study proposal?

It would be very difficult, if not impossible, to successfully conduct this study since randomly sampled people cannot be required to participate in a clinical trial.