Finish Hypothesis Testing + Intro to Confidence Intervals

Lecture 22

Dr. Elijah Meyer

Duke University
STA 199 - Summer 2023

2023-11-14

Checklist

It’s wonderful to be back!

– Clone ae-22

– Keep up with Slack

– Draft Report Due November 15

– Exam-2 released November 16

— Cumulative with a focus on content post Exam-1

Warm Up

Last Time:

\(\mu_{setosa} - \mu_{versi} = 0\)

\(\mu_{setosa} - \mu_{versi} \neq 0\)

Warm Up

– How was one observation on this null distribution created?

– What is the p-value? How do we interpret it?

– Decision?

– Conclusion?

Note: The sample size for each species was 50.

Warm Up: R-Code

When creating null distributions, we have used the following code. Explain the differences between each code chunk.

class_data |> 
  specify(response = correct_guess, success = "Correct") |>
  hypothesize(null = "point", p = .5) |> #fill in the blank
  generate(reps = 1000, type = "draw") |> #fill in the blank
  calculate(stat = "prop") #fill in the blank

## VS 

iris_filter |>
  specify(response = Sepal.Length, explanatory = Species) |>
  hypothesize(null = "independence") |>
  generate(reps = 1000, type = "permute") |>
  calculate(stat = "diff in means", order = c("setosa", "versicolor"))

Significance level

– \(\alpha\)

– “Fixed level testing”

– acts as a threshold for making decisions and conclusions

Significance level

– p-value > \(\alpha\) -> Fail to reject the null hypothesis

– p-value < \(\alpha\) -> Reject the null hypothesis

Significance level

– Set before the research study

– Set = the probability of a type 1 error

Where a type 1 error is incorrectly rejecting the null hypothesis

Issues with fixed level testing include:

– \(\alpha\) = 0.05 vs a p-value of 0.049…. vs 0.051

Scope of Inference

What we often write at the end of research reports

– Who can we generalize our results to?

– What type of relationship (causal vs association)

Scope of Inference

\(\mu_{setosa} - \mu_{versi} = 0\)

\(\mu_{setosa} - \mu_{versi} \neq 0\)

p-value < 0.001

Did we take a random sample?

Do we have random assignment?

Scope of Inference

– Do we have a random sample?

— Yes -> means our sample is representative -> apply our results to the entire population

— No -> apply results to our sample or a similar sample

Scope of Inference

– Do we have random assignment?

— Yes -> even out confounding variables across groups -> causal inference

— No -> confounding variables may be present -> association

Confidence Intervals

– are a range of plausible values that our population parameter could be

– we make confidence intervals when we want to ESTIMATE

Confidence Intervals

By the end of class, we will

– know how to calculate confidence intervals using bootstrap methods

– understand what bootstrap methods are

– understand how the level of confidence influences our confidence interval

– understand how to interpret a confidence interval in the context of the problem