Data science enhances people’s decision making. Doctors and researchers are making critical decisions every day. Therefore, it is absolutely necessary for those people to have some basic knowledge of data science. This series aims to help people that are around medical field to enhance their data science skills.

We will work with a health related database the famous “Pima Indians Diabetes Database”. It was generously donated by Vincent Sigillito from Johns Hopkins University. Please find further information regarding the dataset here.

This is the sixth part of the series and it aims to cover partially the subject of Inferential statistics.
Researchers rarely have the capability of testing many patients,or experimenting a new treatment to many patients, therefore making inferences out of a sample is a necessary skill to have. This is where inferential statistics comes into play.
In more detail, in this part we will go through the hypothesis testing for Student’s t-distribution (Student’s t-test), which may be the most used test you will need to apply, since in most cases the standard deviation σ of the population is not known. We will cover the one-sample t-test and two-sample t-test(both with equal and unequal variance). If you are not aware of what are the mentioned distributions please go here to acquire the necessary background.

Before proceeding, it might be helpful to look over the help pages for the t.test.

Please run the code below in order to load the data set and transform it into a proper data frame format:

url <- "https://archive.ics.uci.edu/ml/machine-learning-databases/pima-indians-diabetes/pima-indians-diabetes.data"
data <- read.table(url, fileEncoding="UTF-8", sep=",")
names <- c('preg', 'plas', 'pres', 'skin', 'test', 'mass', 'pedi', 'age', 'class')
colnames(data) <- names
data <- data[-which(data$mass ==0),]

Answers to the exercises are available here.

If you obtained a different (correct) answer than those listed on the solutions page, please feel free to post your answer as a comment on that page.

Exercise 1

Suppose that we take a sample of 25 candidates that tried a diet and they had a average weight of 29 (generate 25 normal distributed samples with mean 29 and standard deviation 4) after the experiment.
Find the t-value.

Exercise 2

Find the p-value.

Exercise 3

Find the 95% confidence interval.

Exercise 4

Apply t-test with Null Hypothesis that the true mean of the sample is equal to the mean of the sample with 5% confidence level.

Exercise 5

Apply t-test with Null Hypothesis that the true mean of the sample is equal to the mean of the population and the alternative that the true mean is less than the mean of the population with 5% confidence level.

Exercise 6

Suppose that we want to compare the current diet with another one. We assume that we test a different diet to a sample of 27 with mass average of 31(generate normal distributed samples with mean 31 and standard deviation of 5). Test whether the two diets are significantly different.
Note that the two distributions have different variances.
hint: This is a two sample hypothesis testing with different variances.

Exercise 7

Test whether the the first diet is more efficient than the second.

Exercise 8

Assume that the second diet has the same variance as the first one. Is it significant different?

Exercise 9

Assume that the second diet has the same variance as the first one. Is it significantly better?

Exercise 10

Suppose that you take a sample of 27 with average mass of 29, and after the diet the average mass is 28(generate the sampled with rnorm(27,average,4)). Are they significant different?
hint: Paired Sample T-Test.