- 2-sample
*t*test compares the means of**two**populations - ANOVA generalizes this to
**more than two**populations - $H_0:\mu_1=\mu_2=\cdots=\mu_I$, i.e., all the
**population**means are the same - $H_a:$
**not all**of the $\mu_i$ are equal - $H_a$ is
**NOT**$\mu_1\neq\mu_2\neq\cdots\neq\mu_I$

- We usually assume that the (population) variances for each group are the same. To check this, use the following rule:
- If the largest std. is less than twice the smallest standard deviation, we can claim that the assumption is valid.

- $R^2=$ Sum of Squares Between Groups / Total Sum of Squares
- $s=\sqrt{\text{Mean Square Within Groups}}$

- If $H_0$ is rejected, we only know that
**some**of the means are different. - To find out the pairs that have significant differences, we need to do the multiple comparisons using
**Bonferroni**method. - P-value less than $\alpha$ means that the corresponding pair has significant difference.

- Data format for ANOVA is similar to 2-sample
*t*test: one column for data, one column for group.

- Input number code for each fish category, e.g. 1 for largemouth bass, 2 for catfish, 3 for trout, 4 for walleye.