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.

  1. symmetric unimodal distribution histogram, this(2)
  2. multi-modal distribution(more than one peak point), this (3)
  3. right skewed unimodal distribution, this 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: (a) rightly skewed - the median would be a better representation of the data while the IQR would show the direction of the data.

  1. symmetrical - the mean and sd should be sufficient to represent the data.

  2. leftly skewed - the median and IQR should be enough to measure the variability in the data.

  3. symmetric - 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] 0.8823529
## [1] 0.6521739

  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 (a) According to the plot, survival is dependent on the patient got a transplant because the patient who got transplant are more survival days than the patients who did not get transplant. (b) According to the box plots the median of the treatment group have a higher survival time.. Also, the Q1 and Q3 values are higher in the treatment group showing a dependent relationship, or strong efficacy, in heart translates and survival rates. (c)[1] 0.8823529 - 88.24% of the control group died [1] 0.6521739 - 65.22% of the treatment group died.

  1. What are the claims being tested? The null hypothesis is that treatment will not affect the survival rate of patients. The alternative hypothesis is that treatment will improve the survival rate of patients.
  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 ____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_or 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? The simulation graph suggests that there are less examples where the fraction is 0.23., what is a very rare event so we can reject null hypothesis and conclude, having heart transplant affect on the survivability.

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