Extra: CA2

Overview

Coverage: L10-L20, P4-P6

This is a document to supplement the document for CA1. Keep in mind that CA2 covers all lectures and practicals. Please work through CA1_extra_R.qmd again as well.

Note that there are often many possible paths to the correct solution.

library(tidyverse); theme_set(theme_classic())
library(readxl)
library(cowplot)
set.seed(2026)

Air quality

The airquality dataset is available in R upon start-up, and gives daily values of mean ozone, solar radiation, mean wind speed, and max temperature at LaGuardia Airport in New York City (?airquality).

Assume we do not know the population parameters, but wish to estimate the probability of observing our sample mean under the hypothesis that \(\mu = 11 mph\). What is the T statistic?

Under this hypothesized \(\mu\), what is the probability of collecting a sample with a mean further from \(\mu\) than in this dataset?

Iris

The iris dataset is available in R upon start-up.

Research question: What is the difference in mean sepal area among Iris setosa, I. versicolor, and I. virginica?

What type of variables are the response and predictor variables?

What analysis is appropriate for this question?

For this question, is Species a fixed effect or a random effect?

Calculate Sepal.Area

Create a new column in the data.frame iris called Sepal.Area.

Create a histogram of Sepal.Area, showing the distribution for each species using the same limits for the x-axis.

Option 1. ggplot:

Option 2. Base R:

Perform the analysis

Perform the analysis and assess the assumptions.

What assumptions of this ANOVA can you evaluate with these plots? Do they appear reasonable?

What is the null hypothesis?

What is the probability of the observed difference in group means if the null hypothesis were true?

Follow up analysis

What are the means and 95% confidence intervals for each species?

What is the best interpretation of these 95% confidence intervals?

What are the effect sizes and 95% confidence intervals? That is, which species are meaningfully different from one another and how confident are we in those differences?

Iris (reprise)

Research question: For the species I. virginica, what is the effect of sepal width on petal width?

What type of variables are the response and predictor variables?

What analysis is appropriate for this question?

Create the appropriate data.frame

Create a new dataframe containing all columns in iris, but only rows where the species is virginica.

Create a scatterplot of the relationship, with the predictor variable on the x-axis and the response variable on the y-axis.

Option 1. ggplot:

Option 2. Base R:

Perform the analysis

Perform the analysis and check the assumptions.

What assumptions of this linear regression can you evaluate with these plots? Do they appear reasonable?

What is the null hypothesis?

What would the null hypothesis look like on the scatter plot above?

What is the probability of the observed change in petal width per unit increase in sepal width if the null hypothesis were true?

Follow up analysis

To three significant figures, what is the best estimate of the change in petal width per unit increase in sepal width, and what are the 95% confidence intervals?

Create a scatter plot of the relationship and add the best fit line from the linear regression with 95% confidence intervals.

NMDS, PCA

You will not be asked to produce any R code relating to Practical 6, so I am not including extra R practice here. You should nevertheless be familiar with the concepts, confident in the steps involved, and comfortable interpreting the output.