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: The boxplot is created. We can see that 57 lies below the lower whisker while the upper whisker is at the max value of 94. The median is described by the thick line inside the box at 78.5 while Q1 is described by the lower boundary of the box and Q3 by the upper boundary of the box.

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

We can match the histograms and the boxplots using the scale. The histogram in figure(a) is symmetric and matches up nicely with the boxplot in 2 which also displays symmetry in its composition. The histogram in center (b) matches with the boxplot in 3 while the histogram in figure (c) matches nicely with the boxplot in figure 1. This histogram is right-skewed and the boxplot in (1) has several outliers beyond the upper whisker depicting the 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.

ANSWER: The housing prices are positively skewed, the bulk of the house prices are below $1million. Per the description, there are also quite a few outliers with house prices more than $6million. When the data is skewed, the median and the IQR best represent the typical observation and the variability as they are unaffected by outliers.

  1. 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.

ANSWER: Again right-skewed per explanation above, very few outlier prices beyond the $1.2mn mark. Hence, median and IQR would be appropriate metrics to use.

  1. 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.

ANSWER: Again right-skewed per description. Most students don’t drink but a few do excessively. Median and IQR should be used.

  1. 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: Again right-skewed per description. Few executives earn high salaries.

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.

ANSWER: The treatment box plot height or width is twice that of the control group which means that patients in the treatment group have higher survival rates than patients in the control group. Hence, survival rate is the dependent variable whereas transplants is the explanatory variable.

  1. What do the box plots below suggest about the efficacy (effectiveness) of the heart transplant treatment.

ANSWER: The heart transplant program is definitely effective as survival rates are higher with the treatment group.

  1. What proportion of patients in the treatment group and what proportion of patients in the control group died?

ANSWER: The proportion of patients in the treatment group who died is 45/69 = 65.22% The proportion of patients in the control group who died is 30/34 = 88.24%

  1. One approach for investigating whether or not the treatment is effective is to use a randomization technique.
  1. What are the claims being tested?

ANSWER: A randomized experiment can be conducted to test the inference of causality. The claim in this case is that patients who received a transplant increased their survival rates.

  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 equal or greater___. 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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