Example Is the difference between average GPAs in the 7 schools at UOP significant?

How compare means? Can do 21 two-sample t tests. Problem:

Need to extend two-sample procedures into an overall test.

GOAL: Compare two or more population means.

NUMERICAL SUMMARIES: Sample means and standard deviations for each group.

GRAPHICAL SUMMARIES: Side by side boxplots or stemplots

Example Want to compare the average gas mileage of standard four-wheel drive pickup trucks manufactured by Chevrolet, Dodge, and Ford. An experiment is designed in which five vehicles of each type are randomly and independently selected from the population of four-shell drive trucks. Each vehicle is driven in a stationary position for the equivalent of 500 miles. The miles per gallon is computed.

Response variable= type=

Explantory variable= type=

Notation Have I groups, n_i observations in each group, i=1,...,I

x_ij is the j^th observation in group i, N=total sample size

INFERENCE Are the observed differences in the sample means enough to conclude that the population means are different, or are the observed differences what we might see by chance?

Hypotheses H₀

(if we find a difference we don’t know exactly which means were different and would need to do follow-up analysis).

test statistic = variability between the group means

variability within the groups

Variation within the samples

=mean square for error (MSE)

Between sample variability=variability in sample means

overall mean = = (n₁₁+n₂₂+n₃₃)/N

MSG = Sn_i(_i- )²/I-1 = mean square for groups

Test Statistic =F₀= MSG/MSE to compare the variabilities

Follows approximately an F distribution

If H₀ is true, these are equal and F₀ is about 1

If H₀ is not true, MSG is bigger than expect by chance, F₀ large

Use Table E to find the p-value, specifying 2 degrees of freedom

Numerator = I-1 Denominator=N-I

MTB > oneway c1=c2 (observations in c1, subscripts in c2)

- Populations are independent (check data collection description)

- Populations are Normal (check nscores of residuals)

- All of the populations have the same variance (check for ratio of largest standard deviation to smallest standard deviation<2)

Interaction - Does the effect of one explanatory variable on the response depend on the value of a second explanatory variable?