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.

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

Answer 2

Histogram A has few outliers, and it is a nearly normal distribution. There is some symmetry in the distribution. The best match for Histogram A is Boxplot 2. Boxplot 2 is the most normally distributed boxplot. Histogram B has no outliers and there is a somewhat even distribution among the bins. The closest match to Histogram B is Boxplot C. There are no outliers in boxplot C and it is a very widely distributed plot. Histogram C is very right skewed. This distribution is best matched by Boxplot A, which has all of its outliers in the positive direction. The boxplot reinforces the right or positive skew.

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 3

  1. This distribution would be right skewed. The median would be a better representation of the distribution because there is a heavy amount of outliers well above the lowest 75% of the population. Variability would be better measured using IQR because the outliers above $6 million would have a lower impact on the variability than standard deviation would.
  2. This is a symmetric distribution. The distribution is very compact and the outliers have little impact on the distribution. Mean would be the best observation of the data, and variability would be best measured by standard deviation. The less variable the distribution, the more useful standard deviation and mean become.
  3. This distribution is heavily right skewed. The mean and median are located to the right of the distribution. Median and IQR would be the best measures for typical observation and variability analysis.
  4. This distribution would likely be right skewed because the salaries are much higher with a few high earners, which would shift the mean and median to the right. Median and IQR would again be the best measures for analysis of this data. Because the outliers have a high impact on the average salary numbers and there are no outliers on the low end, median is the best method for analysis.

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?

Answer 4:

  1. The mosaic plot indicates that survival is dependent on treatment. Because the experimental group has a higher survival rate than the control group, it can be concluded that survival is dependent on treatment.
  2. The box plots suggest that the heart transplant treatment dramatically increases the survival time for patients.
  3. 15/23 of the treatment group died, which is just over 65% of the members of the treatment group. 15/17 of the control group died, which is just over 88%.
library(openintro)
data("heart_transplant")
heart_transplant
controldead <- subset(heart_transplant, survived == 'dead' & transplant == 'control')
control <- subset(heart_transplant, transplant == 'control')
deadpct <- as.numeric(nrow(controldead))/as.numeric(nrow(control))
deadpct
## [1] 0.8823529
treatdead <- subset(heart_transplant, survived == 'dead' & transplant == 'treatment')
treat <- subset(heart_transplant, transplant == 'treatment')
deadtrtpct <- as.numeric(nrow(treatdead))/as.numeric(nrow(treat))
deadtrtpct
## [1] 0.6521739
deadpct - deadtrtpct
## [1] 0.230179
  1. The claims being tested are that heart transplants increase the possibility for longer lifespan.
  2. 28 75 69 34 0 0.23018
  3. The simulated results show that there is no conclusive evidence that the transplants successfully resolve the issue. Most of the data is centered around the -.05 to 0.12 range, but there is a non-negligible amount of outliers on the left end of the distribution. This distribution is left skewed, so it could be argued that the treatment program actually doesn’t improve the odds of survival. The results from this study were inconclusive.

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