Statistical Methods Notes

Math 222 - Spring 2026

Week 1 Notes

Mon, Jan 12

Today we went over the course syllabus and talked about making R-markdown files in Rstudio. We started the following lab in class, I recommend finishing the second half on your own. I also recommend installing Rstudio on your own laptop (it’s free).

• Lab: Working with R and Rstudio

Wed, Jan 14

Today we reviewed populations and samples. We started with a famous example of a bad sample.

• Slides: Literary Digest election poll 1936

Then we reviewed population parameters, sample statistics, and sampling frames. The difference between a sample statistic and a population parameter is called the sample error.
There are two sources of sample error:

1. Bias. Can be caused by a non-representative sample (sample bias) or by measurement errors, non-response, or biased questions (non-sample bias). The only way to avoid sample bias is a simple random sample (SRS) from the whole population.

2. Random error. This is non-systematic error. It tends to get smaller with larger samples.

To summarize: $$\text{Statistics} = \text{Parameters} + \underbrace{\text{Bias } + \text{ Random error}}_\text{Sample error}.$$

We finished with this workshop.

• Workshop: Sampling

Fri, Jan 16

If you find an association between an explanatory variable and a response variable in an observational study, then you can’t say for sure that the explanatory variable is the cause. We say that correlation is not causation because there might be lurking variables that are confounders, that is, they are associated with both the explanatory and response variables and so you can tell what is the true cause.

It turns out that randomized experiments can prove cause and effect because random assignment to treatment groups controls all lurking variables. We also talked about blocking and double-blind experiments.

• Example: 1954 polio vaccine trials

• Workshop: Experiments

We finished by simulating the results of the polio vaccine trials to see if they might just be a random fluke. We wrote this R code in class:

results = c()
trials <- 1000
for (x in 1:trials) {
  simulated.result <- sample(c(0,1), size = 244, replace = TRUE)
  percent <- sum(simulated.result) / 244
  results <- c(results, percent)
}
hist(results)
sum(results < 0.336) / trials

Week 2 Notes

Wed, Jan 21

Today we did a lab about using R to visualize data.

• Lab: High Bridge half marathon (Rmd, html)

You should be able to open this file in your browser, then hit CTRL-A and CTRL-C to select it and copy it so that you can paste it into Rstudio as an R-markdown document.

We had a little trouble with R-markdown on the lab computers.

Fri, Jan 23

Last time we talked about how to visualize data with R. Here are two quick summaries of how to make plots in R:

• Example: Basic plots with R
• Example: Facier plots with ggplot2

After that, we started talking about probability. We review some of the basic rules.

The notation $P(B \, |\, A)$ means “the probability of B given that A happened”. Two events $A$ and $B$ are independent if the probability of $A$ does not depend on whether or not $B$ happens. We did the following examples.

1. If you shuffle a deck of 52 playing cards and then draw two, what is the probability that the second card is an ace if the first card is?

We also talked about tree diagrams (see subsection 3.2.7 from the book) and how to use them to compute probabilities.

2. Based on a study of women in the United States and Germany, there is an 0.8% chance that a woman in her forties has breast cancer. Mammograms are 90% accurate at detecting breast cancer if someone has it. They are also 93% accurate at not detecting cancer in people who don’t have it. If a woman in her forties tests positive for cancer on a mammogram screening, what is the probability that she actually has breast cancer?

3. 5% of men are color blind, but only 0.25% of women are. Find $P( \text{male} \, | \, \text{color-blind})$.

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Week 14 Notes