Chapter 1 - Introduction to Data

Practice: 1.7 (available in R using the data(iris) command), 1.9, 1.23, 1.33, 1.55, 1.69

Graded: 1.8, 1.10, 1.28, 1.36, 1.48, 1.50, 1.56, 1.70 (use the library(openintro); data(heartTr) to load the data)

Start of homework

1.8 Smoking habits of UK residents. A survey was conducted to study the smoking habits of UK residents. Below is a data matrix displaying a portion of the data collected in this survey. Note that “£” stands for British Pounds Sterling, “cig” stands for cigarettes, and “N/A” refers to a missing component of the data.

  1. What does each row of the data matrix represent?
    Each row represents a response to the survey - an observation in the study.
  2. How many participants were included in the survey?
    1691
  3. Indicate whether each variable in the study is numerical or categorical. If numerical, identify as continuous or discrete. If categorical, indicate if the variable is ordinal.
    sex - categorical
    age - numerical, discrete
    marital - categorical
    grossIncome - categorical, ordinal
    smoke - categorical amtWeekends - numerical, discrete
    amtWeekdays - numerical, discrete


    1.10 Cheaters, scope of inference. Exercise 1.5 introduces a study where researchers studying the relationship between honesty, age, and self-control conducted an experiment on 160 children between the ages of 5 and 15. The researchers asked each child to toss a fair coin in private and to record the outcome (white or black) on a paper sheet, and said they would only reward children who report white. Half the students were explicitly told not to cheat and the others were not given any explicit instructions. Di???erences were observed in the cheating rates in the instruction and no instruction groups, as well as some di???erences across children’s characteristics within each group.
    Identify the population of interest and the sample in this study.
    The population of interest sounds like it may be all people - they want to study how age is related to honesty and self-control. Due to the sample of children between the ages of 5 and 15, the results would only be generalizable - assuming a competent study - to kids in that age range.
    Comment on whether or not the results of the study can be generalized to the population, and if the ???ndings of the study can be used to establish causal relationships.
    We don’t know how the 160 children in the study were chosen, and it’s also not clear how that sample was split into the instruction and no instruction groups. This is an experiement with a control and treatment group of sorts. Assuming sampling was ready for study selection and group assignment, we could generalize over the population of children between the ages of 5 and 15.

    1.28 Reading the paper.
    Below are excerpts from two articles published in the NY Times:
    An article titled Risks: Smokers Found More Prone to Dementia states the following: “Researchers analyzed data from 23,123 health plan members who participated in a voluntary exam and health behavior survey from 1978 to 1985, when they were 50-60 years old. 23 years later, about 25% of the group had dementia, including 1,136 with Alzheimer’s disease and 416 with vascular dementia. After adjusting for other factors, the researchers concluded that pack-aday smokers were 37% more likely than nonsmokers to develop dementia, and the risks went up with increased smoking; 44% for one to two packs a day; and twice the risk for more than two packs.” Based on this study, can we conclude that smoking causes dementia later in life? Explain your reasoning.
    No, this is an observational study, so causation cannot be established. Moreover, it’s voluntary, and selection in the study was not random. However, the associated/correlation effects appear to be significant, and it warrants further study.
    Another article titled The School Bully Is Sleepy states the following: “The University of Michigan study, collected survey data from parents on each child’s sleep habits and asked both parents and teachers to assess behavioral concerns. About a third of the students studied were identi???ed by parents or teachers as having problems with disruptive behavior or bullying. The researchers found that children who had behavioral issues and those who were identi???ed as bullies were twice as likely to have shown symptoms of sleep disorders.” A friend of yours who read the article says, “The study shows that sleep disorders lead to bullying in school children.” Is this statement justi???ed? If not, how best can you describe the conclusion that can be drawn from this study?
    No, this is not a justifiable conclusion. The study was observational and may or may not have been randomized. More than with the previous example, these variables could be confouding as well.

    1.36 Exercise and mental health. A researcher is interested in the e???ects of exercise on mental health and he proposes the following study: Use strati???ed 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.
    What type of study is this?
    Experiement
    What are the treatment and control groups in this study?

    Treatment - exercise
    Control - no exercise

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

    Yes, blocking by age buckets


    Does this study make use of blinding?

    Nope, both subjects and researches likely aware of grouping

    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.


    On a surface level, this is a random experiment with random assignment, so the conclusions could be generalized to the 18-55 population.

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

    Based on other known health benefits of exercise, having a group that is told not to exercise could be unethical.



    1.48 Stats scores. Below are the ???nal exam scores of twenty introductory statistics students.

57, 66, 69, 71, 72, 73, 74, 77, 78, 78, 79, 79, 81, 81, 82, 83, 83, 88, 89, 94

Create a box plot of the distribution of these scores. The ???ve number summary provided below may be useful.

Min Q1 Q2 (Median) Q3 Max 57 72.5 78.5 82.5 94
Build the scores vector.

scores <- c(57, 66, 69, 71, 72, 73, 74, 77, 78, 78, 79, 79, 81, 81, 82, 83, 83, 88, 89, 94)


Summarize scores.

summary(scores)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   57.00   72.75   78.50   77.70   82.25   94.00


Box plot.

boxplot(scores)




1.50 Mix-and-match. Describe the distribution in the histograms below and match them to the box plots.

unimodal - matches 2 uniform - matches 3 right skewed - matches 1

1.56
1.56 Distributions and appropriate statistics, Part II . For each of the following, state whether you expect the distribution to be symmetric, right skewed, or left skewed. Also specify whether the mean or median would best represent a typical observation in the data, and whether the variability of observations would be best represented using the standard deviation or IQR. Explain your reasoning.
Housing prices in a country where 25% of the houses cost below $350,000, 50% of the houses cost below $450,000, 75% of the houses cost below $1,000,000 and there are a meaningful number of houses that cost more than $6,000,000.
Due mostly to the meaningful number of house above $6m, the data is right skewed and therefore would best be represented by the median and IQR. The non-trivial number of extremely expensive houses would provide a misleading mean - and the IQR. However, it’s important not to neglect the meaningful number of expensive homes, as they might be indicative of other economic issues that should be examined.


Housing prices in a country where 25% of the houses cost below $300,000, 50% of the houses cost below $600,000, 75% of the houses cost below $900,000 and very few houses that cost more than $1,200,000.
This seems like a great case for mean and standard deviation - this appears to be a symmetric, unimodally distributed housing market.

Number of alcoholic drinks consumed by college students in a given week. Assume that most of these students don’t drink since they are under 21 years old, and only a few drink excessively.
If only a few student under 21 drink execessively, this would likely be a right skewed distribution best represented by the median and IQR. Still, if, say 80% of the drinks are taken by 20% of the students, that asymmetric distribtuion could lead to a misleading depiction of your “average student,” but it’s also probably a big red flag for further study of that high-consuming population.

Annual salaries of the employees at a Fortune 500 company where only a few high level executives earn much higher salaries than the all other employees.
Right skewed, median/IQR. See 1 and 3 above.

1.70 Heart transplants. The Stanford University Heart Transplant Study was conducted to determine whether an experimental heart transplant program increased lifespan. Each patient entering the program was designated an ocial heart transplant candidate, meaning that he was gravely ill and would most likely bene???t from a new heart. Some patients got a transplant and some did not. The variable transplant indicates which group the patients were in; patients in the treatment group got a transplant and those in the control group did not. Another variable called survived was used to indicate whether or not the patient was alive at the end of the study.


Based on the mosaic plot, is survival independent of whether or not the patient got a transplant? Explain your reasoning.

No, suvival is associated with treatment - a transplant in this case. A much higher proportion of the students who got transplants lived.

What do the box plots below suggest about the efficacy (e???ectiveness) of the heart transplant treatment.

While there are outliers in the control group with longer survival times, the patients who received transplants had longer survival times.


What proportion of patients in the treatment group and what proportion of patients in the control group died? 

Treatment_deaths = 45/69
Treatment_deaths 
## [1] 0.6521739


Control

Control_deaths = 30/34
Control_deaths
## [1] 0.8823529


One approach for investigating whether or not the treatment is effective is to use a randomization technique.

i. What are the claims being tested?
ii. The paragraph below describes the set up for such approach, if we were to do it without using statistical software. Fill in the blanks with a number or phrase, whichever is appropriate.

We write alive on 28 cards representing patients who were alive at the end of the study, and 79 dead on cards representing patients who were not. Then, we shuffle these cards and split them into two groups: one group of size 69 representing treatment, and another group of size 34 representing control. We calculate the di???erence between the proportion of dead cards in the treatment and control groups (treatment control) and record this value. We repeat this 100 times to build a distribution centered at 0. Lastly, we calculate the fraction of simulations where the simulated di???erences in proportions are .230. If this fraction is low, we conclude that it is unlikely to have observed such an outcome by chance and that the null hypothesis should be rejected in favor of the alternative.
iii. What do the simulation results shown below suggest about the e???ectiveness of the transplant program?

Only three occurrences showed a simulated difference of .20 or greater, which means such a difference is unlikely to have occurred by chance.