Stats scores. (2.33, p. 78) 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)
boxplot(scores)

Mix-and-match. (2.10, p. 57) Describe the distribution in the histograms below and match them to the box plots.

Below are types of distributions

a - Bi Modal distribution b - Multi Modal distribution c - Bi Modal distribution

a matches to picture 2 b matches to picture 3 c matches to picture 1

Distributions and appropriate statistics, Part II. (2.16, p. 59) 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.

  1. 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.
  2. 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.
  3. 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.
  4. 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.
housing_prices <- c(7000000,8000000,500000,600000,700000,400000,375000,350000,200000,100000)
boxplot(housing_prices)

summary(housing_prices)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  100000  356250  450000 1822500  675000 8000000
housing_prices_1 <- c(1200000,1300000,700000,800000,850000,750000,375000,350000,200000,100000)
boxplot(housing_prices_1)

summary(housing_prices_1)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  100000  356250  725000  662500  837500 1300000
students_drinks <- c(0,0,0,0,0,0,0,0,50,60)
boxplot(students_drinks)

summary(students_drinks)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##       0       0       0      11       0      60
exec_salaries <- c(1500,2350,2212,19000,18000,3200,1310,2667,1789,2345)
boxplot(exec_salaries)

summary(exec_salaries)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    1310    1895    2348    5437    3067   19000

Below are findings based on the summary and plot above

a,c,d : In a 10 value sample, The distribution is right skewed because mean > median for the given values and it has extreme outliers. Median and IQR can be used to describe the center and spread.

b : In a 10 value sample, The distribution is symmetric as there are no extreme outliers and mean ~ median. Mean and SD can be used to describe center and spread.

Heart transplants. (2.26, p. 76) 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. Of the 34 patients in the control group, 30 died. Of the 69 people in the treatment group, 45 died. Another variable called survived was used to indicate whether or not the patient was alive at the end of the study.

  1. Based on the mosaic plot, is survival independent of whether or not the patient got a transplant? Explain your reasoning.
  2. What do the box plots below suggest about the efficacy (effectiveness) of the heart transplant treatment.
  3. What proportion of patients in the treatment group and what proportion of patients in the control group died?
  4. One approach for investigating whether or not the treatment is effective is to use a randomization technique.
  1. What are the claims being tested?
  2. 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.

  1. What do the simulation results shown below suggest about the effectiveness of the transplant program?

\begin{center} \end{center}

  1. Based on the Plot, there are more number of deaths in the treatment group comparing to control group. This illustrates that the survival chance for the treatment group is higher and all patients in the treatment group got transplanted. So the statement “survival is independent of patient getting transplant” is not correct. There seemes to be good number of patients who have got transplanted have survived.

  2. The box plots suggests how soon the patients have been dead in control and treatment group. In Control group the average survival time is < 50 days but in treatment group it is around 250 days. This also tells us that the upper whisker for survival time is around 1500 days and there are several outliers beyond 1500 days as well too. This clearly indicates that the transplant treatment is effective.

  3. Percentage of death in Control group = 88% Percentage of death in Treatment group = 65%

d.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 ____difference in proportion_____. Lastly, we calculate the fraction of simulations where the simulated differences in proportions are > difference percentage (23%)_____. 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.

The simulation results show that the treatment has impact over survival as Null hypothesis (Treatment has no impact ) is rejected.