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.

    a) Each row of the data matrix represents a case

    b) 1691 participants were included in the survey

    c) sex: categorical  
       age: numerical, discrete  
       marital: categorical  
       grossIncome: categorical, ordinal  
       smoke: categorical  
       amtWeekends: numerical 
       amtWeekdays: numerical

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. Differences were observed in the cheating rates in the instruction and no instruction groups, as well as some differences across children’s characteristics within each group.

    a) The population of interest is children aged 5-15 years old

    b) The results of the study cannot be generalized to the population because whether or not the study utilized random assignment is not specified. The findings of the study can be used to establish correlation statements for the sample because the use of random sampling in the study is unknown.

1.28 Reading the paper

  1. 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-a- day 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, based on this study we cannot conclude that smoking causes dementia later in life because the results were based off of voluntary responses. There was no random sampling so the results cannot be generalized.

  2. 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 identified 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 identified 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 justified? If not, how best can you describe the conclusion that can be drawn from this study?  

The data is collected from surveys so this is an obervational study. Therefore, no causal relationship can be made. Only a correlation statement can be made for the sample: there is a correlation between children bullying in school and sleep disorders.

1.36 Exercise and mental health A researcher is interested in the effects of exercise on mental health and he 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.

    a) This study is a randomized experiment

    b) The treatment group is the exercise group and the control group is the non-exercise group

    c) Yes, this study does make use of blocking. Age is the blocking variable.

    d) No, this study does not make use of blinding.

    e) Since the researcher proposed using stratified random sampling, blocking, and random assignment, causal conclusions can be generalized to the population at large.

    f) My reservations are around variability in the type, duration, and intensity of exercise done by participants in the treatment group. The treatment group participants should follow similar exercise plans.

1.48 Stats scores Below are the final 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 five number summary provided below may be useful.
  scores <- c(57, 66, 69, 71, 72, 73, 74, 77, 78, 78, 79, 79, 81, 81, 82, 83, 83, 88, 89, 94)
  summary(scores)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   57.00   72.75   78.50   77.70   82.25   94.00
  boxplot(scores, main = "Stats Scores", horizontal = TRUE)

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

    a) This is a symmetrical bell curve distribution and goes with box plot 2

    b) This is a uniform distribution and goes with box plot 3

    c) This distribution in skewed right and goes with box plot 1

1.56 Distributions and appropriate statistics, Part I 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.

    a) 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.  

    Right skewed because a meaningful number of houses cost more than $6,000,000. The median and IQR would best represent a typical observation in the data because the extreme values have little affect on them. 

    b) 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.  

    Symmetrical because the quartiles are evenly distributed. The mean would best represent a typical observation in the data because the data is symmetric. The variability would be best represented using standard deviation because there are no extreme values

    c) 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.  

    Right skewed because most students don't drink and only a few drink excessively. Median and IQR because they are robust estimates.

    d) 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 because only a few executives earn much higher salaries than all other employees. Median and IQR because they are robust estimates

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 official heart transplant candidate, meaning that he was gravely ill and would most likely benefit 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.

    a) Based on the mosaic plot, survival is not independent of whether or not the patient got a transplant. A greater proportion of patients in the treatment group are alive.

    b) The box plots suggest that the transplant increased survival time. The mean of the control group is around 10 days and the IQR is about 100. Meanwhile, the mean in the treatment group is about 200 days and the IQR is about 1450.

    c) Load data:
library(openintro)
## Please visit openintro.org for free statistics materials
## 
## Attaching package: 'openintro'
## The following objects are masked from 'package:datasets':
## 
##     cars, trees
data(heartTr)
table(heartTr$transplant, heartTr$survived)
##            
##             alive dead
##   control       4   30
##   treatment    24   45

Proportion of patients in the treatment group that died:

45/69
## [1] 0.6521739

Proportion of patients in the control group that died:

30/34
## [1] 0.8823529

Difference in proportions:

(45/69)-(30/34)
## [1] -0.230179
    d) i. Null hypothesis: The variables transplant and survived are independent; receiving a transplant has no effect on lifespan.
          Alternative hypothesis: The variables transplant and survived are not independent; receiving a transplant does have an effect on lifespan.

      ii. We write alive on **28** cards representing patients who were alive at the end of the study, and dead on **75** 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 difference 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 differences in proportions are **-0.2302**. 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. The simulation results suggest that simulated differences in proportions occur at a rate of about 0.02. Therefore, the null hypothesis is rejected in favor of the alternative hypothesis that the variable tranplant and survived are not independent; receiving a transplant does have an effect on lifespan.