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.

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   57.00   72.75   78.50   77.70   82.25   94.00

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

Solution The distribution in the histogram (a) is Bell-Curve shaped with a matching box-plot (2). The distribution in the histogram (b) is uniformly distributed with a matching box-plot (3) The distribution in the histogram (c) is Right Skewed with a matching box-plot (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.

Solution (a) The distribution would be rightly skewed, the mean and sd will be severely impacted by the outliers making median a better representation of the data while the IQR would show the direction of the data.

  1. This distribution will be symetrical since the quartile ranges are similar. The mean and sd should be sufficient to represent the data.

  2. The distribution will be leftly skewed, the median and IQR should be sufficient to measure the variability in the data.

  3. This distribution will be symmetric making the mean and sd a better measure of 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.

Solution The mosaic plot shows that a large percentage of the treatment group survived compared to the control group. There seems to be a dependant relationship between the treatment (transplant) and survival rate.

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

Solution The treatment (transplant) seems to be significantly effective. Although all the patients did not survive, the transplant saved a significant proportion than the control group.

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

Solution Control Group 4 survived 30 dead Total = 34 Proportion Dead: 30/43 = 88%

Treatment group 24 survived 45 dead Total = 69 Proportion of Dead: 45/69 = 65%

  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?

Solution H0: Survival does not depend on Treatment transplant than control transplant H1: Survival depend on Treatment transplant than control transplant

  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.

Solution

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 zero. 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?

Solution Unable to solve this problem as DATA606 package was not found from devtools::install_github(“jbryer/DATA606”)