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.

hist(scores , breaks=20 , border=F , col='blue' , xlab="distribution of scores")

boxplot(scores , xlab="scores" , col='skyblue' , las=2)

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

The data is rightly skewed as many points are beyond the upper wiskers.

The data is have equal distance is IQR and wiskers.

The data is symmetric skewed as data is normally distributed.

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.
The dataset is right skewed. The distance between 1st Quartile and 2nd Quartile is 100,000 while the distance between 2nd and 3rd Quartile is 550,000. Which means the distribution is more spread between 2nd and 3rd Quartile thus the dataset is right skewed. The median would be a better representation of a typical obsevation while the IQR would be a better representation of the variability sicnce there are meaningfull houses which cost more than 6,000,000, such high prices would greatly effect the Standard deviation as well as the mean significantly.
  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.
The distance between 1st Quartile and 2nd Quartile and 3rd Quartile are 300,000 each, whicj shows that the all quartiles have similar spread, thus the data is symmetric. Also since there are few houses which cost more than $1,200,000 the data shoulnt be greaty skewed. Here the mean would be a better representation of the data set, as well as the Standard Deviation becasue the data set is symmetric and there are few observation which shouldnt skew the data too much.
  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.
The distribution is right-skewed since most students don’t consume any alcohol and only a few drink excessively. Median should be used to represent typical observation while IQR to represent variability since the the data is rightly skewed.
  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.
Here we shoudl see similar salaries and the distribution as symmetric.Mean should be used with standard deviation to represent variability in the data set.

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.
The survival is not independent because there is a difference in the survival rate of those who got the transplant vs those who didnot. If it was independent, then the transplant should have not changed the survival ratio.
  1. What do the box plots below suggest about the efficacy (effectiveness) of the heart transplant treatment.
There treatment has a significant effect on the days of surviaval. All the quartiles are significatly higher than the control group.
  1. What proportion of patients in the treatment group and what proportion of patients in the control group died?
Control group= 30/34 died = .882 or 88.2% died
Treatment group= 45/69 died = .652 or 65.2% died
  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?
H0 Experimental heart transplant and survical rate is Independent.
H1 Experimental heart transplant and survival rate are dependent
  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 .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?

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We can see in the simulation that the our proportion of .23 is towards the end of the tails which means that it is very unique and didnot happen by chance. Thus we can reject the H0 as there is strong evidence that the experimental heart transplant does increase lifespan time and survival rate.