Question

An amusement park studied methods for decreasing the waiting time (minutes) for rides by loading and unloading riders more efficiently. Two alternative loading/unloading methods have been proposed. To account for potential differences due to the type of ride and the possible interaction between the method of loading and unloading and the type of ride, a factorial experiment was designed. Use the following data to test for any significant effect due to the loading and unloading method, the type of ride, and interaction. Use  = .05. Factor A is method of loading and unloading; Factor B is the type of ride. Type of Ride Roller Coaster Screaming Demon Long Flume Method 1 43 56 47 45 48 43 Method 2 50 51 52 52 47 48 Set up the ANOVA table (to 2 decimal, if necessary). Round p-value to four decimal places. Source of Variation Sum of Squares Degrees of Freedom Mean Square F p-value Factor A Factor B Interaction Error Total The p-value for Factor A is Selectless than .01between .01 and .025between .025 and .05between .05 and .10greater than .10Item 21 What is your conclusion with respect to Factor A? SelectFactor A is significantFactor A is not significantItem 22 The p-value for Factor B is Selectless than .01between .01 and .025between .025 and .05between .05 and .10greater than .10Item 23 What is your conclusion with respect to Factor B? SelectFactor B is significantFactor B is not significantItem 24 The p-value for the interaction of factors A and B is Selectless than .01between .01 and .025between .025 and .05between .05 and .10greater than .10Item 25 What is your conclusion with respect to the interaction of Factors A and B? SelectThe interaction of factors A and B is significantThe interaction of factors A and B is not significantItem 26 What is your recommendation to the amusement park? SelectUse method 1; it has a lower sample mean waiting time and is the best methodWithhold judgment; take a larger sample before making a final decisionSince method is not a significant factor, use either loading and unloading methodItem 27

Answer 1

The Summary Statistics are as below

A	2
B	3
n	2
N	12

Total	Roller	Demon	Flume	Total Row
Method1	88	114	90	292
Method2	102	98	100	300
Total Column	190	212	190	592

(1) (Overall Total)² / N = (592)²/12 = 29205.23

(2) Sum of Squares of Individual Observations = 43² + 56² + .....+ 48² = 29454

(3) SUM (Row Total)² / (b*n); b * n = 3 * 2 = 6

(292)²/6 + (300)²/6 = 29210.67

(4) SUM (Column Total)² / (a * n); a * n = 2 * 2 = 4

(190)²/4 + (212)²/4 + (190)²/4 = 29286

(5) SUM(Cell)²/n; n = 2

(88)²/2 + (114)²/2 + (90)²/2 + (102)²/2 + (98)²/2 + (100)²/2 = 29424

____________________________________________________________________

SS A = (3) - (1) = 29210.67 - 29205.23 = 5.33

SS B = (4) - (1) = 29286 - 29205.23 = 80.67

SS AB = (5) + (1) - (3) - (4) = 29424 + 29205.23 - 29210.67 - 29286 = 132.67

SS Error = (2) - (5) = 29454 - 29424 = 30

SS Total = 5.33 + 80.67 + 132.67 + 30 = 248.67

df A = a - 1 = 3 - 1 = 2

df B = b - 1 = 2 - 1 = 1

df AB = df A * df B = 2 * 1 = 2

df Error = N - (a * b) = 12 - (3 * 2) = 6

MS A = SS A/df A = 5.33

MS B = SS B/df B = 40.34

MS AB = SS AB/df AB = 66.34

MS Error = SS Error / df Error = 5

F A = MS A/ MS error = 1.07

F B = MS B/ MS error = 8.07

F AB = MS AB/ MS error = 13.27

The Complete ANOVA Table with the p values are as below.

	SS	df	MS	F	P value
A	5.33	1	5.33	1.07	0.3408
B	80.67	2	40.34	8.07	0.0199
AB	132.67	2	66.34	13.27	0.0063
Error	30	6	5
Total	248.67	11

(1) The p value for Factor A is > 0.10

The Factor A is not significant (Since p value is > )

___________________________________________

(2) The p value for Factor B is between 0.01 and 0.05

The Factor A is significant (Since p value is < )

____________________________________________

(3) The p value for Factor A and B is < 0.01

The Factor AB is significant (Since p value is < )

____________________________________________

The Recommendation: Since method is not significant use either method 1 or method 2 for loading and unloading.

_____________________________________________