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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:

(a)Symmetric and unimodal. Match to (2)

(b)Symmetric and Multimodal. Match to (3)

(c)Right skewed and unimodal. Match to (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.

    Answer:It would be right skewed and we need to use median because $6,000,000 will make the mean large. In addition, we need to use IQR.

  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.

    Answer: The distribution will be symmetric because price doesn’t changes much for every percentage level. We can also use mean and Standard deviation because very few houses that cost more than $1,200,000. I don’t think mean and median, and standard deviation and IQR will make many differences.

  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.

    Answer: The distribution will be right skewed because most of the student don’t drink and only a few students drink excessively.Those excessively drinker will make the data mean far away from the 50% level, so we should choose median and IQR.

  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: The distribution will be right skewed as well because only a few high level executives earn much higher salaries.Those higher salary will make the data out of those regular data, so we should choose median and IQR.

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: No. Survival is not independent of those patients who got the transplant because the portion of alive for treatment is much bigger than the portion of alive for control.

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

Answer: The heart transplant treatment save patients. From the box plot, the treatment group have a higher median, Q1 and Q3.

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

proportion of patients in the treatment group died

45/69
## [1] 0.6521739

proportion of patients in the control group died

30/34
## [1] 0.8823529
  1. One approach for investigating whether or not the treatment is effective is to use a randomization technique.

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 (-0.2302). 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: Simulation results shows that the -0.2302 difference only has 3% chance. Such a low probability indicates a rare event.