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

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

Answers:

  1. Symmetric normal distribution. Matches 2.

  2. Multi-modal distribution. Matches 3.

  3. Right-skewed distribution. Matches 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.

Answers:

  1. Distribution is right-skewed since there are a meaningful number of houses that cost more than $6mil. Median should be used for right-skewed distributions, and IQR to represent variability.

  2. Distribution is symmetric since the difference between the first and second, and third and fourth quartiles are similar, so mean should be used, and st.dev. to represent variability.

  3. Distribution is right-skewed since most students don’t consume any alcohol and only a few drink a lot. Median should be used with IQR to represent variability.

  4. Distribution is symmetric since most employees earn similar salaries, so mean should be used with st.dev. to represent variability.

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.

Answers:

  1. Survival is not independent based on the mosaic plot. The dead area of the control group is taller than that of the treatment group. If survival was independent they should be the same height.
  2. The box plots suggest that treatment has a significant effect on survival time, as all quartiles of the treatment group have significantly higher survival time.
  3. Treatment group 45/69 = 0.65 = 65% Control group 30/34 = 0.88 = 88%
  1. What are the claims being tested?

Whether or not the treatment is effective in increasing a patient’s survival time.

  1. 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 ____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 _0.23__. 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.

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

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Answer:

The graph shows that there are few simulations where the fraction is 0.23, so we were unlikely to have observed this outcome by chance. So, the null hypothesis should be rejected and we can conclude that survival is impacted by having a heart transplant. m