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 700 1,700 2,280 1,400 1,650 1,670 850 1,630 1,740 Slant 1,430 1,850 2,000 1,740 1,700 1,510 1,240 1,650 2,480 Rear-end 700 860 1,650 1,250 1,550 1,650 970 1,250 1,240 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 a b (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 a b (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 Slant Rear-End Total Two-Factor ANOVA with Replication Source SS df MS F p-value Factor 1 (Angle) Factor 2 (Vehicle) Interaction Error Total (c) Using α = 0.05, choose the correct statements. The main effects of angle and vehicle are significant, but there is not a significant interaction effect. The main effect of vehicle is significant; however, there is no significant effect from angle or interaction between angle and vehicle. The main effect of angle is significant; however, there is no significant effect from vehicle or interaction between angle and vehicle.

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 700 1,700 2,280 1,400 1,650 1,670 850 1,630 1,740 Slant 1,430 1,850 2,000 1,740 1,700 1,510 1,240 1,650 2,480 Rear-end 700 860 1,650 1,250 1,550 1,650 970 1,250 1,240 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 a b (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 a b (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 Slant Rear-End Total Two-Factor ANOVA with Replication Source SS df MS F p-value Factor 1 (Angle) Factor 2 (Vehicle) Interaction Error Total (c) Using α = 0.05, choose the correct statements. The main effects of angle and vehicle are significant, but there is not a significant interaction effect. The main effect of vehicle is significant; however, there is no significant effect from angle or interaction between angle and vehicle. The main effect of angle is significant; however, there is no significant effect from vehicle or interaction between angle and vehicle.

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

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ISBN:9780079039897

Author:Carter

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5 mph Collision Damage ($)
Crash Type	Goliath	Varmint	Weasel
Head-On	700	1,700	2,280
	1,400	1,650	1,670
	850	1,630	1,740
Slant	1,430	1,850	2,000
	1,740	1,700	1,510
	1,240	1,650	2,480
Rear-end	700	860	1,650
	1,250	1,550	1,650
	970	1,250	1,240

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

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

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

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

a.	H₀: Not all the AB_jk are equal to zero	⇐⇐ there is an interaction effect
	H₁: All the AB_jk are equal to zero	⇐⇐ there is no interaction effect
b.	H₀: All the AB_jk are equal to zero	⇐⇐ there is no interaction effect
	H₁: Not all the AB_jk 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
Slant
Rear-End
Total

Two-Factor ANOVA with Replication
Source	SS	df	MS	F	p-value
Factor 1 (Angle)
Factor 2 (Vehicle)
Interaction
Error
Total

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

The main effects of angle and vehicle are significant, but there is not a significant interaction effect.
The main effect of vehicle is significant; however, there is no significant effect from angle or interaction between angle and vehicle.
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 1 decimal place.)

Post hoc analysis for Factor 1:

Tukey simultaneous comparison t-values (d.f. = 18)
	Rear-End	Head-On	Slant

Rear-End
Head-On
Slant
Critical values for experimentwise error rate:
	0.05
	0.01

Post hoc analysis for Factor 2:

Tukey simultaneous comparison t-values (d.f. = 18)
	Goliath	Varmint	Weasel

Goliath
Varmint
Weasel
critical values for experimentwise error rate:
	0.05
	0.01

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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