Wrong By Design

STAT 20: Introduction to Probability and Statistics

Agenda

• Concept Questions
• Problem Set 15

Concept Questions

Instead of constructing a confidence interval to learn about the parameter, we could assert the value of a parameter and see whether it is consistent with the data using a hypothesis test. Say you are interested in testing whether there is a clear majority opinion of support or opposition to the project.

What are the null and alternative hypotheses?

01:00

The null is the p = .5 and the alternative is that p != .5.

This brings up a good discussion of one- and two-tailed tests. When students share their answers, you can draw a picture of a null distribution on the board with the observed statistic as a vertical line and consider how the p-value calculation would change depending on the alternative hypothesis.

This also brings up a discussion of the link between hypothesis tests and confidence intervals. Below the picture of the null distribution on the board, you could draw a bootstrap distribution centered on the vertical line of the observed statistic. In the HT setting, you consider the location of the statistic relative to the null distribution. In making a decision with a CI, you consider the location of the parameter relative to the bootstrap distribution (or more generally, the sampling distribution of the statistic).

library(tidyverse)
library(infer)
library(stat20data)

ppk <- ppk |>
  mutate(support_before = Q18_words %in% c("Somewhat support", 
                                          "Strongly support",
                                          "Very strongly support"))

library(tidyverse)
library(infer)
library(stat20data)

ppk <- ppk |>
  mutate(support_before = Q18_words %in% c("Somewhat support", 
                                          "Strongly support",
                                          "Very strongly support"))
obs_stat <- ppk |>
  specify(response = support_before,
          success = "TRUE") |>
  calculate(stat = "prop")

library(tidyverse)
library(infer)
library(stat20data)

ppk <- ppk |>
  mutate(support_before = Q18_words %in% c("Somewhat support", 
                                          "Strongly support",
                                          "Very strongly support"))
obs_stat <- ppk |>
  specify(response = support_before,
          success = "TRUE") |>
  calculate(stat = "prop")
obs_stat

Response: support_before (factor)
# A tibble: 1 × 1
   stat
  <dbl>
1 0.339

null <- ppk |>
  specify(response = support_before,
          success = "TRUE") |>
  hypothesize(null = "point", p = .5) |>
  generate(reps = 500, type = "draw") |>
  calculate(stat = "prop")

null <- ppk |>
  specify(response = support_before,
          success = "TRUE") |>
  hypothesize(null = "point", p = .5) |>
  generate(reps = 500, type = "draw") |>
  calculate(stat = "prop")
null

Response: support_before (factor)
Null Hypothesis: point
# A tibble: 500 × 2
   replicate  stat
   <fct>     <dbl>
 1 1         0.516
 2 2         0.5  
 3 3         0.481
 4 4         0.504
 5 5         0.498
 6 6         0.493
 7 7         0.493
 8 8         0.492
 9 9         0.501
10 10        0.489
# ℹ 490 more rows

null <- ppk |>
  specify(response = support_before,
          success = "TRUE") |>
  hypothesize(null = "point", p = .5) |>
  generate(reps = 500, type = "draw") |>
  calculate(stat = "prop")
visualize(null) +
  shade_p_value(obs_stat, direction = "both")

What would a Type I error be in this context?

Answer: Concluding that there is a clear majority when in fact there is an even split.

01:00

What would a Type II error be in this context?

Answer: Concluding that there is an even split when in fact there is a clear majority.

After the previous question, students should get this quite easily.

Problem Set 15

30:00