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.

Ans:

  1. Histogram <-> (2) BoxPlot

This (a) Histogram is Gaussian (Normal) distribution and corresponds to (2) Boxplot with a mean of 60

  1. Histogram <-> (3) Boxplot

This (b) Histogram is a uniform distribution and correponds to (3) Boxplot. There are no outliers and values are distributed evenly throughout the range from 0 to 100.

  1. Histogram <-> (1) Boxplot

Histogram (c) is a skewed right and it corresponds to boxplot (1). The majority of values fall in the lower end of the range between about 0.8 and 2 with a lot of outliers beyond the upper Q3. Also it is lower bounded by 0

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.

Ans:

Q1 = 350K

Q2 = 450K (Median)

Q3 = 1M

IQR = 650K

Upper fence= Q3+1.5IQR = 1.975M

Lower fence= Q1-1.5IQR = (625K) which is NOT possible as home prices cannot fall below zero

Problem states -> Meaningful no. of homes greater than 6M

  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.

Ans:

Q1 = 300K

Q2 = 600K (Median)

Q3 = 900k

IQR = 600K

Upper Fence= Q3+1.5IQR = 1.8M

Lower Fence= Q1-1.5IQR = (600K) which is NOT possible as home prices cannot fall below zero

Problem states -> very few home prices above 1.2M

  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 of college drinkers is right skewed since most students don’t drink and underage and only a few drink excessively, so the majority of values are at the lower end of the range.

A typical distribution should be described by the median and the variability would best be described by the IQR since they are not too affected by outliers

  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.

The distribution of typical salaries is mostly normal (symmetric) since most employees’ salaries are bunch up together quite closely with only a few high-end executives earning outrages amounts; CEO, COO, CFO. All the ones with 3-letter acronyms attached to their titles

The graph would showed quite a symmetric distribution set but to describe salaries:

This is not easy, if you use mean and standard deviation, the average salaries is skewed toward the high end because its pulled by the high salaried executives and is misleading. So its better to use median and IQR to represent salaries as its less susceptible to outlier events.

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.

Yes, looks like more survived than died when provided Treatment

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

It showed that 50% of the patients who received treatment had longer survival time, alluding to the fact that treated patients (heart transplants) survived longer

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

alive dead

0.3478261 0.6521739

Dead = 65% in the treatment group

alive      dead

0.1176471 0.8823529

Dead = 88% in Control Group

##    id acceptyear age survived survtime prior transplant wait
## 8  38         70  41     dead        5    no  treatment    5
## 16 95         73  40     dead       16    no  treatment    2
## 18  3         68  54     dead       16    no  treatment    1
## 19 74         72  29     dead       17    no  treatment    5
## 23 20         69  55     dead       28    no  treatment    1
## 24 70         72  52     dead       30    no  treatment    5
## 
## alive  dead 
##    24    45
## 
##     alive      dead 
## 0.3478261 0.6521739
##   id acceptyear age survived survtime prior transplant wait
## 1 15         68  53     dead        1    no    control   NA
## 2 43         70  43     dead        2    no    control   NA
## 3 61         71  52     dead        2    no    control   NA
## 4 75         72  52     dead        2    no    control   NA
## 5  6         68  54     dead        3    no    control   NA
## 6 42         70  36     dead        3    no    control   NA
## 
## alive  dead 
##     4    30
## 
##     alive      dead 
## 0.1176471 0.8823529
  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?

That the treatment and the outcomes are independent of each other or Not indepent

  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 _0.03______. 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.

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RPubs Link: http://rpubs.com/ssufian/525919