Question

1. Question 1: A researcher wants to know whether supplements affect how long the undead can go without requiring human blood. To investigate this, she took one sample of vampires and one sample of zombies. Half the vampires and half the zombies received a placebo, while the other half of vampires and zombies received a supplement. She then recorded how many days the vampires and zombies could go without feeding. The goal of the study was to investigate three different things: whether there was an effect of supplement on number of days; whether there was an effect of type of undead (zombie or vampire) on number of days; and whether there was an interaction between the two factors. Conduct the appropriate ANOVA for these data using α = 0.05 and calculate partial − η² as a measure of effect size. Assume that all assumptions for the test are met. (See table below for sample data.)

Factor B: supplement

	Supplement	Placebo
	M =17.38	M =11.85	Trow =380
Zombie	T =226	T =154	nrow =26
	SS =563.08	SS =369.69	Mrow =14.62
	n =13	n =13	SDrow =6.73
	M =26.38	M =21.46	Trow =622
Vampire	T =343	T =279	nrow =26
	SS =847.08	SS =189.23	Mrow =23.92
	n =13	n =13	SDrow =6.91
	Tcol =569	Tcol =433	N =52
	ncol =26	ncol =26	G =1002
	Mcol =21.88 SDcol =8.8	Mcol =16.65 SDcol =6.81	ΣX2 =22760

Answer 1

N = 52

Replications, r = 13

ΣX = 1002

(ΣX)² =1004004

ΣX² = 22760

SSA = Σ((ΣXⱼ)²/nⱼ) - (ΣX)²/N = (380²/26 + 622²/26) - 1004004/52 = 1126.2308

SSB = Σ((ΣXᵢ)²/nᵢ) - (ΣX)²/N = (569²/26 + 433²/26) - 1004004/52 = 355.6923

SSBN = Σ((ΣX)²/n) - (ΣX)²/N = 1483.1538

SSAxB = SSBN - SSA - SSB = 1.2308

SSW = SST - SSA - SSB - SSAxB = 1969.0769

SST = ΣX² - (ΣX)²/N = 22760 - 1004004/52 = 3452.2308

dfA = a - 1 = 1

dfB = b-1 = 1

dfAxB = (a-1)*(b-1) = 1

dfW = ab(r-1) = 48

dfT = N-1 = 51

MSA = SSA/dfA = 1126.2308/1 = 1126.2308

MSB = SSB/dfB = 355.6923/1 = 355.6923

MSAxB = SSAxB/dfAxB = 1.2308/1 = 1.2308

MSW = SSW/dfW = 1969.0769/48 = 41.0224

F for Factor A = MSA/MSW = 27.4540

p-value for Factor A = F.DIST.RT(27.454, 1, 48) = 0.0000

Critical value for Factor A = F.INV.RT(0.05, 1, 48) = 4.0427

F for Factor B = MSB/MSW = 8.6707

p-value for Factor B = F.DIST.RT(8.6707, 1, 48) = 0.0050

Critical value for Factor B = F.INV.RT(0.05, 1, 48) = 4.0427

F for interaction = MSAxB/MSW = 0.0300

p-value for Interaction = F.DIST.RT(0.03, 1, 48) = 0.8632

Critical value for Interaction = F.INV.RT(0.05, 1, 48) = 4.0427

ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Treatment	1483.1538	3
Factor A	1126.2308	1	1126.2308	27.4540	0.0000	4.0427
Factor B	355.6923	1	355.6923	8.6707	0.0050	4.0427
Interaction	1.2308	1	1.2308	0.0300	0.8632	4.0427
Within	1969.0769	48	41.0224
Total	3452.2308	51

For Factor A:

As p = 0.000< 0.05, we reject the null hypothesis.

There was an effect of type of undead (zombie or vampire) on number of days

For Factor B:

As p = 0.005< 0.05, we reject the null hypothesis.

There was an effect of supplement on number of days.

For interaction:

As p = 0.863 > 0.05, we fail to reject the null hypothesis.

There is no interaction between the two factors

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η² for factor A:

η²A = SSA /(SST - SSB - SSAxB) = 1126.2308/(3452.2308 - 355.6923 - 1.2308) = 0.3639 = 36.39%

η² for factor B:

η²B = SSB /(SST - SSA - SSAxB) = 355.6923/(3452.2308 - 1126.2308 - 1.2308) = 0.153 = 15.3%

η² for Interaction:

η²AxB = SSAxB/(SST- SSA - SSB) = 1.2308/(3452.2308 - 1126.2308 - 355.6923) = 0.0006 = 0.06%