Question

Twenty-seven high-school seniors decided to take part in an investigation of the special “exam preparation” books that purportedly help one get ready for college entrance examinations. The group divided itself into three groups on a purely random basis. Two of the groups used the books, each group selecting a different book. The third group did not use the books. Listed below are the obtained entrance exam scores. Perform the ANOVA using the 5% level of significance and interpret your results.

BOOK I: 532 455 440 620 560 522 517 520 510

BOOK II: 540 570 520 620 660 605 602 590

NO BOOK: 380 470 441 487 420 390 450 510 430 560

Answer 1

	A	B	C
count, ni =	9	8	10
mean , x̅ i =	519.556	588.38	453.80
std. dev., si =	53.010	44.638	54.980
sample variances, si^2 =	2810.028	1992.554	3022.844
total sum	4676	4707	4538		13921	(grand sum)
grand mean , x̅̅ =	Σni*x̅i/Σni =	515.59
square of deviation of sample mean from grand mean,( x̅ - x̅̅)²	15.705	5297.279	3818.324
					TOTAL
SS(between)= SSB = Σn( x̅ - x̅̅)² =	141.346	42378.231	38183.245		80702.82
SS(within ) = SSW = Σ(n-1)s² =	22480.222	13947.875	27205.600		63633.697

no. of treatment , k =   3
df between = k-1 =    2
N = Σn =   27
df within = N-k =   24
  
mean square between groups , MSB = SSB/k-1 =    40351.4106
  
mean square within groups , MSW = SSW/N-k =    2651.4041
  
F-stat = MSB/MSW =    15.2189

anova table
	SS	df	MS	F	p-value	F-critical
Between:	80702.82	2	40351.41	15.22	0.0001	3.40
Within:	63633.70	24	2651.40
Total:	144336.52	26
					α =	0.05

conclusion :    p-value<α , reject null hypothesis            
so, means are different

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