Review: Situation - compare several population means

H₀: m₁=m₂=…=m_I I=number of groups/populations

H_a: at least one m_i differs

Test statistic = F = variability between group means df=I-1

variability within groups df=N-I

Sampling distribution of this test statistic follows an F(I-1,N-I) distribution. P-values for this distribution are found in Table E.

• Populations are independent (check data collection description)
• Populations are Normal
• All of the populations have the same variance

Example: Comparing truck gas mileage

F= 2.50, p=.124, conclusion: Fail to reject H₀

MTB> oneway c1=c2 c3(data in c1, subscripts in c2, stores residuals in c3)

To check normality: Could do a plot for each group (e.g. truck). Analyzing residuals lets us examine normality of the combined data from all groups.

Normal Prob Plot of residuals

To check equal variances:

Will consider this assumption reasonably met if

Largest standard deviation < 2

Smallest standard dev

Truck SDs: .35, .25, .44

Example Paula can take either Main St or High St to work, and she wants to know if one route gets her to work faster?

Response variable= type=

Explanatory variable= type=

She randomly chooses different routes for her drive to work for the next four weeks, makes a total of ten trips with each route.

Main St: 26.0 24.2 26.5 24.1 24.2 19.9 16.7 20.7 20.5 17.2 High St: 18.8 20.0 22.7 22.3 21.2 17.4 17.3 12.7 15.6 12.0

₁=22 min s₁=3.49 min

₂=18 min s₂=3.74 min

Example 2 Dispuva, Feely, and Hwang (1997) conducted a study to see if there is a difference in car speeds depending on the gender of the driver or the color of the car. They took a sample of 100 cars traveling west on Alpine Street between 12:30 and 2:30. At one tree, one observer began a stopwatch the moment the car passed the tree. At a second tree, 234' away, a second observer lowered his arm the moment the car passed to signal the first observer to stop the time. A third observer recorded the gender of the driver (to the best of his abilities). One-way ANOVA:

Here is the output from a pooled t-test (assuming equal variances and using s_p² for SE(₁ - ₂) ):

• Compare the t statistic and its p-value with the F statistic and its p-value. Is there a relationship between t₀ and F₀? The p-values?
• What general observations can you make about the pooled t-test and the F-test when there are only two groups to compare?
• Verify the calculation of "POOLED STDEV" and indicate how you can find this value from the ANOVA table.
• Would you consider these data as independent SRSs? Explain.