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.

Answer

Creating a boxplot one with whole scores

Creating a boxplot one with the five number summary

boxplot(fivenum(scores))

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

Answer

  1. Symmetric unimodal distribution and matches to boxplot(2)
  2. Uniform multimodal distribution and matches to boxplot(3)
  3. Right skewed bimodal distribution and matches to boxplot(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.

Answer

  1. Distribution here is right skewed. Median would represent a typical observation in the data. The variability of observations would be best represented with IQR because there are a good number of expensive houses and this would increase the mean and sd.
  2. Data is symmetrically distributed. Since the data is evenly distributed mean would be used for represent a observation and standard deviation to represent variability.
  3. The data is right skewed. Median would represent a typical observation in the data. The variability of observations would be best represented with IQR because large number of students don’t drink.
  4. The data is right skewed. Median would represent a typical observation in the data. The variability of observations would be best represented with IQR because the high level executive salary is much higher than the average salary

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}

Answer

## [1] "(a) There is difference of  23.0179028132992 % chances of survival between treatment and control group. So survival is not independent of whether or not the patient got a transplant"
## [1] "(b) The box plots show that the median of the treatment group have a higher survival time of about 100 days. Also, the Q1 and Q3 values are higher in the treament group suggesting a strong efficacy in heart transplants and survival rates."
## [1] "(c) Portion of patients died in treatment group is  65.2173913043478 %. Portion of patients died in control group is  88.2352941176471 %"
## [1] "(d) (i) The claims being tested are whether an experimental heart transplant program increases lifespan."
## [1] "(d) (ii) We write *alive* on ____28______ cards representing patients who were \nalive at the end of the study, and *dead* on ____75_____ cards \nrepresenting patients who were not. Then, we shuffle these cards and split them \ninto two groups: one group of size __69_______ representing treatment, and \nanother group of size ____34______ representing control. We calculate the \ndifference between the proportion of *dead* cards in the treatment and \ncontrol groups (treatment - control) and record this value. We repeat this 100 \ntimes to build a distribution centered at ___0______. Lastly, we calculate \nthe fraction of simulations where the simulated differences in proportions are \n____ -0.230179028132992 _____. If this fraction is low, we conclude that it is unlikely to \nhave observed such an outcome by chance and that the null hypothesis should \nbe rejected in favor of the alternative."
## [1] "d (iii) The outcomes are unlikely to be independent of the treatment, or be due to chance."