Tutorial Exercises: $t$-test and non-parametric equivalents

11.11. Tutorial Exercises: $t$-test and non-parametric equivalents#

Here are some more exercises on comparing means using the t-test and non-parametric equivalents

These exercises are very similar to what you did in the t-test and Mann-Whitney/Wilcoxon examples so in most cases you will be able to copy and adapt code and text from the examples.

Set up Python libraries#

As usual, run the code cell below to import the relevant Python libraries

# Set-up Python libraries - you need to run this but you don't need to change it
import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stats
import pandas 
import seaborn as sns

1. Whose peaches are heavier?#

There should be a picture of some peaches here

Mr Robinson’s juice factory buys peaches from farmers by the tray. Each tray contains 50 peaches. Farmer McDonald claims that this is unfair as his peaches are juicier and therefore weigh more than the peaches of his rival, Mr McGregor.

Mr Robinson weighs eight trays of Farmer McDonald’s peaches and 8 trays of Mr McGregor’s peaches. The weights, in kilograms are given in the file peaches.csv found at https://raw.githubusercontent.com/jillxoreilly/StatsCourseBook/main/data/peaches.csv

Throughout the coures you have been loading files directly from my Github repository using pandas.read_csv using ready made code - below you will need to copy and modify an example to read the file peaches.csv from that same repository

Investigate whether McDonald’s claim is justified by testing for a difference in weight between McDonald and McGregor’s peaches. Use both a parametric and non-parametric test.

a) Load the data into a Pandas dataframe

#Your code here to load the data - check previous examples if unsure how to do this

b) Plot the data and comment.

A Kernel desity estimate plot (to show the distribution) and rug plot (to show individual data points) would be a good choice here. You should comment on whether the data appear to be Normally distriubted and hence the suitability of the t-test.

# your text here to plot the data

d) We can assume (based on the Central Limit Theorem) that these data points are normally distributed. Explain why.

< Your text here explaining why the data should be Normal according to the CLT >

e) Conduct a t-test to test Farmer McDonald’s claim

State your hypotheses
State relevant descriptive statistics
Carry out the test using the ubilt in function from scipy.stats with appropriate option choices
State your conclusions

< Your answer here! You will need to add additional cells >

f) Conduct an appropriate non-parametric test of Farmer McDonald’s claim

State your hypotheses
State relevant descriptive statistics
Carry out the test using the ubilt in function from scipy.stats with appropriate option choices
State your conclusions

2. IQ and vitamins#

There should be a picture of some vitamin pills here

The VitalVit company claim that after taking their VitalVit supplement, IQ is increased.

They run a trial in which 22 participants complete a baseline IQ test, then take VitalVit for six weeks, then complete another IQ test.

The participants’ scores can be found in the data file VitalVit.csv at https://raw.githubusercontent.com/jillxoreilly/StatsCourseBook/main/data/vitalVit.csv

a. What kind of design is this.

< your answer here >

b. What are the advantages and possible disadvantages of this type of design? Should the company have done something different or additional to rule out confounding factors?

< your answer here >

c. Load the data into a Pandas dataframe

#Your code here to load the data - check previous examples if unsure how to do this

d. Plot the data and comment. A scatterplot would be a good choice as these are paired data. You could add the line of equality (line x=y) to the graph so we can see whether most people score higer on the IQ test before or after taking VitalVit

# Your code here for a scatter plot. 
# You can find an example in the workbook on paired sample t-test

IQ scores are normally distributed by design (the tests are designed to yeild a normal distribution of scores). Therefore we should be able to use a t-test to compare the scores from before and after taking VitalVit. We will carry out both the (parametric) t-test and a suitable non-parametric test

e) Conduct a t-test to test VitalVit’s claim

State your hypotheses
State relevant descriptive statistics
Carry out the test using the ubilt in function from scipy.stats with appropriate option choices
State your conclusions

< Your answer here! You will need to add additional cells >

f) Conduct a suitable non-parametric test to test VitalVit’s claim

State your hypotheses
State relevant descriptive statistics
Carry out the test using the ubilt in function from scipy.stats with appropriate option choices
State your conclusions

< Your answer here. You will probably need to add more cells. >

3. Who has the tallest students?#

A student from Lonsdale college claims that Lonsdale students are taller than students from Beaufort college.

Heights of 30 randomly selected male undergraduates from each college are found in the file heightsCollege.csv found at https://raw.githubusercontent.com/jillxoreilly/StatsCourseBook/main/data/heightsCollege.csv

Test the student’s hypothesis using a t-test (this is justified as heights are generally normally distributed) and write up your report as if for a scientific publication. Your report should include the following elements:

A plot of the data to show the data distribution
The relevant descriptive statistics
The results of the t-test
A conclusion

You can use the write-up sections of the t-test example notebooks as a model

#Your code here to load the data - check previous examples if unsure how to do this