Question

​Seventy-eight people participated in a study of weight loss with a specified diet. The following table lists the amount of weight loss for randomly selected participants when they followed the​ diet, catagorized by age group.

Age Group	16​ - 25	26​ - 35	36​ - 45	46​ - 55
Data	4.2 7.0 5.4 3.4 2.4	8.5 1.9 5.3 3.3 1.1 2.7 0.9	1.7 4.5 9.2 0 3.5 ​-2.1	3.0 0.6 3.6 5.8 0.7

Is there evidence that different age groups have different amounts of weight​ loss?

Provide answers to the following question.

a. State the null and the alternative hypothesis for the test at the α=0.01 level.

b. Name the​ test, list the test statistic and the​ p-value for the test.

c. State the conclusion of the test in context of the problem.

Answer 1

a)

Ho: µ1=µ2=µ3
H1: not all means are equal

b)

One Way ANOVA

count, ni =	5	7	6	5
mean , x̅ i =	4.480	3.39	2.80	2.74
std. dev., si =	1.787	2.705	3.934	2.17
sample variances, si^2 =	3.192	7.318	15.480	4.728
total sum	22.4	23.7	16.8	13.7	76.6	(grand sum)
grand mean , x̅̅ =	Σni*x̅i/Σni =	3.33

square of deviation of sample mean from grand mean,( x̅ - x̅̅)²	1.322	0.003	0.281	0.349
					TOTAL
SS(between)= SSB = Σn( x̅ - x̅̅)² =	6.608	0.021	1.688	1.743	10.06012
SS(within ) = SSW = Σ(n-1)s² =	12.768	43.909	77.400	18.912	152.9886

no. of treatment , k =   4
df between = k-1 =    3
N = Σn =   23
df within = N-k =   19
  
mean square between groups , MSB = SSB/k-1 =    3.3534
  
mean square within groups , MSW = SSW/N-k =    8.0520
  
F-stat = MSB/MSW =    0.4165
P value =   0.7432

	SS	df	MS	F	p-value
Between:	10.06	3	3.35	0.42	0.7432
Within:	152.99	19	8.05
Total:	163.05	22
					α =	0.01
			conclusion :	p-value>α , do not reject null hypothesis

c)

There is not sufficient evidence that means weight loss are different for age groups

THANKS

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