Question

A consumer advocate examines whether the longevity of car batteries (measured in years) is affected by the brand name (factor A) and whether or not the car is kept in a garage (factor B). Interaction is suspected. The results are shown in the accompanying table.

	Brand Name of Battery
Kept in Garage?	A	B	C
Yes	8, 7, 7	7, 6, 7	8, 9, 10
No	6, 7, 6	5, 6, 4	5, 7, 7

Click here for the Excel Data File

a-1. Construct an ANOVA table. (Round intermediate calculations to at least 4 decimal places. Round "SS", "MS" to 4 decimal places and "F", "p-value" to 3 decimal places.)

a-2. At the 5% significance level, Is there interaction between the brand name and whether a car is garaged?

• Yes since the p-value for interaction is less than the significance level.
• Yes since the p-value for interaction is greater than the significance level.
• No since the p-value for interaction is greater than the significance level.
• No since the p-value for interaction is less than the significance level.

b. At the 5% significance level, can you conclude that the average battery lives differ by brand name?

• No since the p-value for brand name is less than the significance level.
• Yes since the p-value for brand name is less than the significance level.
• No since the p-value for brand name is less than the significance level.
• No since the p-value for brand name is greater than the significance level.

c. At the 5% significance level, can you conclude that the average battery lives differ depending on whether a car is garaged?

• Yes since the p-value for garage is greater than the significance level.
• Yes since the p-value for garage is less than the significance level.
• No since the p-value for garage is less than the significance level.
• No since the p-value for garage is less than the significance level.

Answer 1

Source of variation	Degree of Freedom	Sum of Squares	Mean Sum of Squares	F Value
Between levels of Factor A	p-1
Between levels of Factor B	q-1
Interaction between A and B	(p-1)(q-1)
Error	pq(m-1)
Total	mpq-1

Above is the ANOVA table for 2 way analysis of data with m observation per cell.

Where denotes mean of row, denotes mean of column, denotes mean of row and column, refers to mean of all observation and i = 1, 2, 3, ...,p & j = 1, 2, 3, ...., q

In our case, Factor B has 2 levels, ie p = 2 and factor A has 3 levels, q = 3.

Calculation,

required values:

	1	2	3
1	7.3333	6.6667	9	7.6667
2	6.3333	5	6.3333	5.8889
	6.8333	5.8333	7.6667

= 6.778

a1) ANOVA Table

Source of variation	Degree of Freedom	Sum of Squares	Mean Sum of Squares	F Value
Between levels of Factor A	1		14.222	19.692
Between levels of Factor B	2		5.0555	6.991
Interaction between A and B	2		1.0555	1.462
Error	12		0.7222
Total	17

a2) No, since the p-value for interaction (0.730) is more than the significance level (0.05)

b) Yes, since the p-value for brand name (0.004) is less than the significance level 0.05.

c) Yes, since the p-value for garage (0.000) is less than the significance level 0.05.