Fin
Math 58B - Spring 2026
Random
| Random sampling | Random assignment | |
|---|---|---|
| Issue | How to obtain observational units to be studied | How observational units come to be in groups to be compared |
| Goal | Select a sample that is representative of the population in all respects | Produce groups that are as similar as possible in all respects, except for the treatment being assigned |
| Conclusion | Generalize results from sample to population | Draw cause-and-effect conclusion, if difference in response between groups is statistically significant |
Inference
Probability vs. Statistics
Randomization
Break the relationship between the explanatory and response variable to generate a null sampling distribution.
\(\rightarrow\) the null sampling distribution sets up the p-value calculation.
Bootstrapping
Generate new samples from the original data to generate a sampling distribution (where the null is not necessarily true!)
\(\rightarrow\) the sampling distribution leads to a confidence interval for the parameter of interest.
Mathematical modeling
Can generate an approximate null sampling distribution and plain sampling distribution using mathematical tools.
\(\rightarrow\) leads to both p-values for hypothesis testing and also to confidence intervals.
p-value
p-value is the probability of the observed data or more extreme given the null hypothesis is true.
Errors
- What errors can be made in inferential conclusions?
- How is Type I error (\(\alpha\)) control important?
- How is power important?
- How is sample size important?
Model Building
AI
Statisticians are vital to the beginning and the end of the process.
- What questions to ask?
- How were the data generated?
- Can causation be claimed?
- Can you generalize to the population?
- What is the context within which the analysis is taking place?