Statistics 2

In-Class Exercise 12

Yu-You Liou

Shih Chien University

2026-09-16

Question 1

Tukey Test

Use the teaching-method data from Exercise 11 and the model fit <- aov(scores ~ method, data = dta).

  1. Run TukeyHSD(fit).
  2. List the pairs of methods whose mean scores differ significantly at \alpha=0.05.
  3. Explain in one sentence how you read the result from the confidence interval of each pair.

Question 2

Why Not Repeated t Tests?

  1. With three groups, how many pairwise t tests would be needed?
  2. If each test is carried out at \alpha=0.05, the probability of making at least one Type I error is 1-(1-0.05)^{m}, where m is the number of tests. Compute it in R.
  3. Explain in one sentence why ANOVA followed by the Tukey test is preferred.

Question 3

Two-Way ANOVA

A study measured plant growth under two levels of light and two types of fertiliser, with three plants in each combination.

growth <- c(12, 14, 13, 18, 20, 19,
            15, 16, 14, 25, 27, 26)
light  <- factor(rep(c("Low", "High"), each = 6))
fert   <- factor(rep(rep(c("A", "B"), each = 3), 2))
plants <- data.frame(light, fert, growth)
  1. Fit the model with aov(growth ~ light * fert, data = plants) and read summary().
  2. Report whether each main effect is significant at \alpha=0.05.
  3. Is the interaction significant? Explain in one sentence what an interaction would mean here.