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 for the Scores distribution is as below:

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

Answer: The Distribution in the histograms and their match to the Boxplots are:
Histogram a data is symmetric and matches to the the Boxplot 2.
Histogram b data is symmetric and matches to the the Boxplot 3.
Histogram c data is Right Skewed and matches to the the Boxplot 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: The Distribution will be Right Skewed; The Median will represent best over here due to 50% of the observation being below $550,000 and a good number of observation being outliers at more then $6,000,000; will make the average/mean distorted. Thus the variability of observations would be best represented using the 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; The Mean will represent best over here due to 50% of the observation being below $600,000 and the upper whisker being double of it at $1,200,000 and only few outliers above the upper whisker. Thus the variability of observations would be best represented using the Standard Deviation.
  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 and the Median along with IQR will better represent the dataset here; This is due to the few heavy drinkers influencing the outliers.
  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 and the Median along with IQR will better represent the dataset here; This is due to the few high salary executives influencing the outliers.

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: Based on the mosaic plot, the survival rate is higher (34.78%) with a Transplant against a lower survival rate (11.77%) with NO transplant; this shows that there is dependency between transplant and survival. But the sample of control group is small enough to make it a valid reasoning for survival being dependent on having a Transplant.
  2. What do the box plots below suggest about the efficacy (effectiveness) of the heart transplant treatment.
    Answer: The Boxplot suggest more people will benefit from the heart transplant treatment.
  3. What proportion of patients in the treatment group and what proportion of patients in the control group died?
    Answer: 0.65 Proportion of patients in treatment group died; and 0.88 Proportion of patients in control group died.
  4. 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: Claim of having the transplant has higher survival rate.
  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 11 cards representing patients who were alive at the end of the study, and dead on 9 cards representing patients who were not. Then, we shuffle these cards and split them into two groups: one group of size 13 representing treatment, and another group of size 7 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 [((24/69) - (4/34)) i.e. 0.17]. 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: The Result show most of the data are below the study proportion of 0.17 thus we must reject the NULL hypotheses.

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