The city of Seattle, WA has an open data portal that includes pets registered in the city. For each registered pet, we have information on the pet’s name and species. The following visualization plots the proportion of dogs with a given name versus the proportion of cats with the same name. The 20 most common cat and dog names are displayed. The diagonal line on the plot is the \(x = y\) line; if a name appeared on this line, the name’s popularity would be exactly the same for dogs and cats.
Are these data collected as part of an experiment or an observational study?
What is the most common dog name? What is the most common cat name?
What names are more common for cats than dogs?
Is the relationship between the two variables positive or negative? What does this mean in context of the data?
Solution.
Observational study.
Dog: Lucy. Cat: Luna.
Oliver and Lily.
Positive, as the popularity of a name for dogs increases, so does the popularity of that name for cats.
2.Stressed out, Part II.
In a study evaluating the relationship between stress and muscle cramps, half the subjects are randomly assigned to be exposed to increased stress by being placed into an elevator that falls rapidly and stops abruptly and the other half are left at no or baseline stress.
What type of study is this?
Can this study be used to conclude a causal relationship between increased stress and muscle cramps?
3.Chia seeds and weight loss.
Chia Pets -- those terra-cotta figurines that sprout fuzzy green hair -- made the chia plant a household name. But chia has gained an entirely new reputation as a diet supplement. In one 2009 study, a team of researchers recruited 38 men and divided them randomly into two groups: treatment or control. They also recruited 38 women, and they randomly placed half of these participants into the treatment group and the other half into the control group. One group was given 25 grams of chia seeds twice a day, and the other was given a placebo. The subjects volunteered to be a part of the study. After 12 weeks, the scientists found no significant difference between the groups in appetite or weight loss. 1
What are the experimental and control treatments in this study?
Has blocking been used in this study? If so, what is the blocking variable?
Has blinding been used in this study
Comment on whether or not we can make a causal statement, and indicate whether or not we can generalize the conclusion to the population at large.
Solution.
Experiment.
Treatment: 25 grams of chia seeds twice a day, control: placebo.
Yes, gender.
Yes, single blind since the patients were blinded to the treatment they received.
Since this is an experiment, we can make a causal statement. However, since the sample is not random, the causal statement cannot be generalized to the population at large.
4.City council survey.
A city council has requested a household survey be conducted in a suburban area of their city. The area is broken into many distinct and unique neighborhoods, some including large homes, some with only apartments, and others a diverse mixture of housing structures. For each part below, identify the sampling methods described, and describe the statistical pros and cons of the method in the city’s context.
Randomly sample 200 households from the city.
Divide the city into 20 neighborhoods, and sample 10 households from each neighborhood.
Divide the city into 20 neighborhoods, randomly sample 3 neighborhoods, and then sample all households from those 3 neighborhoods.
Divide the city into 20 neighborhoods, randomly sample 8 neighborhoods, and then randomly sample 50 households from those neighborhoods.
Sample the 200 households closest to the city council offices.
5.Flawed reasoning.
Identify the flaw(s) in reasoning in the following scenarios. Explain what the individuals in the study should have done differently if they wanted to make such strong conclusions.
Students at an elementary school are given a questionnaire that they are asked 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.
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.
An orthopedist administers a questionnaire to 30 of his patients who do not have any joint problems and finds that 20 of them regularly go running. He concludes that running decreases the risk of joint problems.
Solution.
Non-responders may have a different response to this question, e.g. parents who returned the surveys likely don’t have difficulty spending time with their children.
It is unlikely that the women who were reached at the same address 3 years later are a random sample. These missing responders are probably renters (as opposed to homeowners) which means that they might be in a lower socio-economic status than the respondents.
There is no control group in this study, this is an observational study, and there may be confounding variables, e.g. these people may go running because they are generally healthier and/or do other exercises.
6.Income and education in US counties.
The scatterplot below shows the relationship between per capita income (in thousands of dollars) and percent of population with a bachelor’s degree in 3,143 in the US in 2010.
What are the explanatory and response variables?
Describe the relationship between the two variables. Make sure to discuss unusual observations, if any.
Can we conclude that having a bachelor’s degree increases one’s income?
7.Eat better, feel better.
In a public health study on the effects of consumption of fruits and vegetables on psychological well-being in young adults, participants were randomly assigned to three groups: (1) diet-as-usual, (2) an ecological momentary intervention involving text message reminders to increase their fruits and vegetable consumption plus a voucher to purchase them, or (3) a fruit and vegetable intervention in which participants were given two additional daily servings of fresh fruits and vegetables to consume on top of their normal diet. Participants were asked to take a nightly survey on their smartphones. Participants were student volunteers at the University of Otago, New Zealand. At the end of the 14-day study, only participants in the third group showed improvements to their psychological well-being acrossthe 14-days relative to the other groups. 2
Tamlin S Conner et al. “Let them eat fruit! The effect of fruit and vegetable consumption on psychological well-being in young adults: A randomized controlled trial”. In: PloS one 12.2 (2017), e0171206.
What type of study is this?
Identify the explanatory and response variables.
Comment on whether the results of the study can be generalized to the population.
Comment on whether the results of the study can be used to establish causal relationships.
A newspaper article reporting on the study states, “The results of this study provide proof that giving young adults fresh fruits and vegetables to eat can have psychological benefits, even over a brief period of time.” How would you suggest revising this statement so that it can be supported by the study?
Solution.
Randomized controlled experiment.
Explanatory: treatment group (categorical, with 3 levels). Response variable: Psychological well-being.
No, because the participants were volunteers.
Yes, because it was an experiment.
The statement should say “evidence” instead of “proof”.
8.Screens, teens, and psychologial well-being.
In a study of three nationally representative large-scale data sets from Ireland, the United States, and the United Kingdom (\(n = 17,247\)), teenagers between the ages of 12 to 15 were asked to keep a diary of their screen time and answer questions about how they felt or acted. The answers to these questions were then used to compute a psychological well-being score. Additional data were collected and included in the analysis, such as each child’s sex and age, and on the mother’s education, ethnicity, psychological distress, and employment. The study concluded that there is little clear-cut evidence that screen time decreases adolescent well-being. 3
Comment on whether the results of the study can be generalized to the population, and why.
Comment on whether the results of the study can be used to establish causal relationships.
9.Stanford Open Policing.
The Stanford Open Policing project gathers, analyzes, and releases records from traffic stops by law enforcement agencies across the United States. Their goal is to help researchers, journalists, and policymakers investigate and improve interactions between police and thepublic. 4
The following is an excerpt from a summary table created based off of the data collected as part of this project.
% of stopped
County
State
Driver’s race
No. of stops per year
cars searched
drivers arrested
Apaice County
Arizona
Black
266
0.08
0.02
Apaice County
Arizona
Hispanic
1008
0.05
0.02
Apaice County
Arizona
White
6322
0.02
0.01
Cochise County
Arizona
Black
1169
0.05
0.01
Cochise County
Arizona
Hispanic
9453
0.04
0.01
Cochise County
Arizona
White
10826
0.02
0.01
...
...
...
...
...
...
Wood County
Wisconsin
Black
16
0.24
0.10
Wood County
Wisconsin
Hispanic
27
0.04
0.03
Wood County
Wisconsin
White
1157
0.03
0.03
What variables were collected on each individual traffic stop in order to create to the summary table above?
State whether each variable is numerical or categorical. If numerical, state whether it is continuous or discrete. If categorical, state whether it is ordinal or not.
Suppose we wanted to evaluate whether vehicle search rates are different for drivers of different races. In this analysis, which variable would be the response variable and which variable would be the explanatory variable?
Solution.
County, state, driver’s race, whether the car was searched or not, and whether the driver was arrested or not.
All categorical, non-ordinal.
Response: whether the car was searched or not. Explanatory: race of the driver.
10.Space launches.
The following summary table shows the number of space launches in the US by the type of launching agency and the outcome of the launch (success or failure). 5
What variables were collected on each launch in order to create to the summary table above?
State whether each variable is numerical or categorical. If numerical, state whether it is continuous or discrete. If categorical, state whether it is ordinal or not.
Suppose we wanted to study how the success rate of launches vary between launching agencies and over time. In this analysis, which variable would be the response variable and which variable would be the explanatory variable?
