Question 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.58
num sex age marital grossIncome smoke amtWeekends amtWeekdays
1 Female 42 Single Under £2,600 Yes 12 cig/day 12 cig/day
2 Male 44 Single £10,400 to £15,600 No N/A N/A
3 Male 53 Married Above £36,400 Yes 6 cig/day 6 cig/day
. . . . . . . .
. . . . . . . .
. . . . . . . .
1691 Male 40 Single £2,600 to £5,200 Yes 8 cig/day 8 cig/day
(a) What does each row of the data matrix represent?
(b) How many participants were included in the survey?
(c) 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.
Answer 1.8
  1. a record or a separate case of a UK resident who participated in the survey
  2. 1691
  3. as the following table:
sex age marital grossIncome smoke amtWeekends amtWeekdays
categorical discrete categorical categorical ordinal categorical discrete discrete
Question 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) Identify the population of interest and the sample in this study.
(b) Comment on whether or not the results of the study can be generalized to the population, and if the findings of the study can be used to establish causal relationships.
Answer 1.10
  1. as the following table:
study_variables population sample
count Undetermined 160
age between 5 and 15 between 5 and 15
sex Male and Female Male and Female
  1. The questions 1.5 and 1.10 do not give more details on how much randomness was applied in the selection of participants into the study and into instructing and non-instructing groups. But the process of the study can be generalized to the population (it could have been better if the count of the children was greater than 160) and the findings have causal relationships.
Question 1.28 - Reading the paper. Below are excerpts from two articles published in the NY Times:
(a) 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.
(b) 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?
Answer 1.28
  1. Actually, no. This study is not an experimental study and smoking has no causal relationships with dementia later in life. It’s observational and limited to health plan membership holders which excludes non-membership holders who can be smokers with dementia.
  2. No, that statement is not justified. The friend’s statement is trying to form a causal relationship in a reversed way between the causal variables. First, the study is observational and no causal relationships can be concluded. Second, the direction goes from bullying to sleep disorders and not the other way around, but we may find a bi-directional relationship between the behavorial issues and the sleep disorders.
Question 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) What type of study is this?
(b) What are the treatment and control groups in this study?
(c) Does this study make use of blocking? If so, what is the blocking variable?
(d) Does this study make use of blinding?
(e) 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.
(f) Suppose you are given the task of determining if this proposed study should get funding. Would you have any reservations about the study proposal?
Answer 1.36
  1. An experimental study
  2. The treatment group is: exercise twice a week, and the control group is: not exercising
  3. Yes. As seen, the study is sampled by proportions of age. Age is the blocking variable
  4. The question has no clear indication about this, but it says that there’s randomly assignment. If the researcher or if the groups operator doesn’t know or is not aware of which participant is exercising or not (kept anonymous) until after the end of the study, then yes.
  5. What’s the population size? Yes, it can be generalized since it’s an randomized experimental study with clear causal relationship between exercising and mental health and with unclear state of blinding during the study.
  6. Yes, there’s a need of reviewing the study proposal with the following:
Reasons
Why the use of stratified random sampling? why not using cluster that can be cheaper for sampling from a population?
Length of study time
Blinding bias: does the researcher knows about who’s exercising and who’s not before the last health exam or after at the end of the study?
Determining the population size
What’s the minimum time of exercising every twice a week?
Question 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:
Min Q1 Q2 (Median) Q3 Max
57 72.5 78.5 82.5 94
Answer 1.48

Question 1.50 -

Answer 1.50
## [1] "--------------------  ----------------------  ------------------------------"
## [2] "(a)                   (b)                     (c)                           "
## [3] "symmetric, unimodal   symmetric, multimodal   skewed to the right, unimodal "
## [4] "2                     3                       1                             "
Question 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.
(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.
(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.
(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.
(d) Annual salaries of the employees at a Fortune 500 company where only a few high level executives earn much higher salaries than all the other employees.
Answer 1.56
  1. dis tribution
    25% < 350000
    50% < 450000
    75% < 1000000
    meaningful_number > 6000000

Right skewed, Median and IQR

  1. dis tribution1
    25% < 300000
    50% < 600000
    75% < 900000
    very_few > 1200000

Skewed to the right, Median and IQR

  1. Since there’s a conditional variable which is 21 and older drink, then it’s skewed to the right, Median and IQR

  2. Since there’s a conditional variable which is few high level executives earn much higher salaries than all the other employee, then it’s skewed to the right, Median and IQR

Question 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. Of the 34 patients in the control group, 30 died. Of the 69 people in the treatment group, 45 died.

(a) Based on the mosaic plot, is survival independent of whether or not the patient got a transplant? Explain your reasoning.
(b) What do the box plots below suggest about the efficacy (effectiveness) of the heart transplant treatment.
(c) What proportion of patients in the treatment group and what proportion of patients in the control group died?
(d) 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 ______ cards representing patients who were alive at
the end of the study, and dead on ______ cards representing patients
who were not. Then, we shuffle these cards and split them into two groups:
one group of size ______ representing treatment, and another group of
size ______ 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 ______. Lastly, we calculate the fraction of simulations where
the simulated differences in proportions are ______. 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 effectiveness of the transplant program?

Answer 1.70
  1. Based on the mosaic plot, the width of the treatment is double than the width of the control concluding that the participants’s number in the treatment is double than the other. Seeing the height of the treatment alive > control alive by about three times, one can conclude that the survival is dependent of whether or not the patient got a transplant.

Using the openintro library and comparing the above with the following, we notice that:

control_alive_count control_dead_count treatment_alive_count treatment_dead_count
4 30 24 45

The total count of the treatment group is 69, almost the double of the total count of the control group: treatment_total = 2 * control_total + 1, treatment_alive_count = 6 * control_alive_count => 6 times better results.

  1. The efficacy (effectiveness) of the heart transplant treatment is definitely helpful and effective and is helping to extend the survival lifetime. From the openintro library, we can prove the box plots treatment effectiveness indications: total_treatment_survival_time > total_control_survival_time, total_treatment_survival_time = ~9.5 * total_control_survival_time.
total_control_survival_time total_treatment_survival_time
1869 17236
  1. total_control_treatment_deaths = control_dead_count + treatment_dead_count control_death_over_total = control_dead_count / total_control_treatment_deaths treatment_death_over_total = treatment_dead_count / total_control_treatment_deaths
control_death_over_total treatment_death_over_total total_control_treatment_deaths
0.4 0.6 75
  1. Whether the transplant has any effect on longevity of survival time or not.
  2. total_alive = 24+4 = 28, total_dead = 30+45 = 75, total_treatment = 24+45 = 69, total_control = 4+30 = 34 => blank_filling_orderly = [ 28, 75, 69, 34, 0, equal or greater ]
  3. Rlating the prargraph in (ii) with the results shown in the image, we can conclude that simulated differences in proportions are observed and show the effectiveness of the transplant treatment.