Question

In a bumper test, three test vehicles of each of three types of autos were crashed into a barrier at 5 mph, and the resulting damage was estimated. Crashes were from three angles: head-on, slanted, and rear-end. The results are shown below. Research questions: Is the mean repair cost affected by crash type and/or vehicle type? Are the observed effects (if any) large enough to be of practical importance (as opposed to statistical significance)?

5 mph Collision Damage ($)
Crash Type	Goliath	Varmint	Weasel
Head-On	750	1,710	2,290
	1,440	1,620	1,630
	850	1,610	1,790
Slant	1,430	1,800	2,090
	1,760	1,750	1,500
	1,220	1,670	2,480
Rear-end	790	850	1,610
	1,230	1,540	1,600
	950	1,230	1,200

  Click here for the Excel Data File

(a-1) Choose the correct row-effect hypotheses.

a.	H0: A1 ≠ A2 ≠ A3 ≠ 0	⇐⇐ Angle means differ
	H1: All the Aj are equal to zero	⇐⇐ Angle means are the same
b.	H0: A1 = A2 = A3 = 0	⇐⇐ Angle means are the same
	H1: Not all the Aj are equal to zero	⇐⇐ Angle means differ

• b

• a

(a-2) Choose the correct column-effect hypotheses.

a.	H0: B1 ≠ B2 ≠ B3 ≠ 0	⇐⇐ Vehicle means differ
	H1: All the Bk are equal to zero	⇐⇐ Vehicle means are the same
b.	H0: B1 = B2 = B3 = 0	⇐⇐ Vehicle means are the same
	H1: Not all the Bk are equal to zero	⇐⇐ Vehicle means differ

• b

• a

(a-3) Choose the correct interaction-effect hypotheses.

a.	H0: Not all the ABjk are equal to zero	⇐⇐ there is an interaction effect
	H1: All the ABjk are equal to zero	⇐⇐ there is no interaction effect
b.	H0: All the ABjk are equal to zero	⇐⇐ there is no interaction effect
	H1: Not all the ABjk are equal to zero	⇐⇐ there is an interaction effect

• a

• b

(b) Fill in the missing data. (Round your table of means values to 1 decimal place, SS and F values to 2 decimal places, MS values to 3 decimal places, and p-values to 4 decimal places.)

Table of Means
	Factor 2 (Vehicle)
Factor 1 (Angle)	Goliath	Varmint	Weasel	Total
Head-On	1013.333	1646.6667	1903.3333
Slant	1470.00	1740.00	2023.33
Rear-End	990.00	1206.6667	1470.000
Total

Two-Factor ANOVA with Replication
Source	SS	df	MS	F	p-value
Factor 1 (Angle)	Not attempted	Not attempted	Not attempted	Not attempted	Not attempted
Factor 2 (Vehicle)	Not attempted	Not attempted	Not attempted	Not attempted	Not attempted
Interaction	Not attempted	Not attempted	Not attempted	Not attempted	Not attempted
Error	Not attempted	Not attempted	Not attempted
Total	Not attempted	Not attempted

(c) Using α = 0.05, choose the correct statement.

• The main effect of vehicle is significant; however, there is no significant effect from angle or interaction between angle and vehicle.

• The main effects of angle and vehicle are significant, but there is not a significant interaction effect.

• The main effect of angle is significant; however, there is no significant effect from vehicle or interaction between angle and vehicle.

(d) Perform Tukey multiple comparison tests. (Input the mean values within the input boxes of the first row and input boxes of the first column. Round your t-values and critical values to 2 decimal places and other answers to 3 decimal places.)

Post hoc analysis for Factor 1:

Tukey simultaneous comparison t-values (d.f. = 18)
		Rear-End	Head-On	Slant
		Not attempted	Not attempted	Not attempted
Rear-End	Not attempted
Head-On	Not attempted	Not attempted
Slant	Not attempted	Not attempted	Not attempted
Critical values for experimentwise error rate:
		0.05	Not attempted
		0.01	Not attempted

Post hoc analysis for Factor 2:

Tukey simultaneous comparison t-values (d.f. = 18)
		Goliath	Varmint	Weasel
		Not attempted	Not attempted	Not attempted
Goliath	Not attempted
Varmint	Not attempted	Not attempted
Weasel	Not attempted	Not attempted	Not attempted
critical values for experimentwise error rate:
		0.05	Not attempted
		0.01	Not attempted

Answer 1

(a-1) correct row-effect hypotheses:

b.	H0: A1 = A2 = A3 = 0	⇐⇐ Angle means are the same
	H1: Not all the Aj are equal to zero	⇐⇐ Angle means differ

(a-2) Choose the correct column-effect hypotheses.

b.	H0: B1 = B2 = B3 = 0	⇐⇐ Vehicle means are the same
	H1: Not all the Bk are equal to zero	⇐⇐ Vehicle means differ

(a-3) Choose the correct interaction-effect hypotheses.

b.	H0: All the ABjk are equal to zero	⇐⇐ there is no interaction effect
	H1: Not all the ABjk are equal to zero	⇐⇐ there is an interaction effect

(b) Fill in the missing data. (Round your table of means values to 1 decimal place, SS and F values to 2 decimal places, MS values to 3 decimal places, and p-values to 4 decimal places.)

Table of Means
	Factor 2 (Vehicle)
Factor 1 (Angle)	Goliath	Varmint	Weasel	Total
Head-On	1013.333	1646.6667	1903.3333	1521.1
Slant	1470.00	1740.00	2023.33	1744.4
Rear-End	990.00	1206.6667	1470.000	1222.2
Total	1157.8	1531.1	1798.9

ANOVA
Source of Variation	SS	df	MS	F	P-value
Factor 1 (Angle)	1235785.19	2	617892.593	6.90	0.0060
Factor 2 (Vehicle)	1866318.52	2	933159.259	10.42	0.0010
Interaction	198814.81	4	49703.704	0.55	0.6981
Error	1612333.33	18	89574.074
Total	4913251.85	26

(c) Using α = 0.05, choose the correct statement.

• The main effects of angle and vehicle are significant, but there is not a significant interaction effect.

(d) Perform Tukey multiple comparison tests. (Input the mean values within the input boxes of the first row and input boxes of the first column. Round your t-values and critical values to 2 decimal places and other answers to 3 decimal places.)

Post hoc analysis for Factor 1:

Tukey simultaneous comparison t-values (d.f. = 18)
		Rear-End	Head-On	Slant
		1222.2	1521.1	1744.4
Rear-End	1222.2
Head-On	1521.1
Slant	1744.4
Critical values for experiment wise error rate:
		0.05	2.55
		0.01	3.32

Post hoc analysis for Factor 2:

Tukey simultaneous comparison t-values (d.f. = 18)
		Goliath	Varmint	Weasel
		1157.8	1531.1	1798.9
Goliath	1157.8
Varmint	1531.1
Weasel	1798.9
critical values for experiment wise error rate:
		0.05	2.55
		0.01	3.32