Question

5. (36 pts) Suppose three samples are selected from three populations as follows: Sample 1: 2, 6 Sample 2: 6, 8, 10 Sample 3: 9, 11, 13, 15 At the α = 0.01 level of significance, determine whether there is evidence of a difference in the population means.

Answer 1

Since there are three samples, we need to use an ANOVA for testing the equality of means.

Null hypothesis:

At least one pair of means are different.

Level of significance:

The given data is presented with the totals for samples and the overall total.

Sample1	Sample 2	Sample3
2	6	9
6	8	11
	10	13
		15
8	24	48	80

We have the number of observations in each sample as

The total umber of observations is =2+3+4=9

We calculate the following :

1). Correction factor

  

2). Total sum of squares:

  

We make use of the table below for individual squares and add them to get the sum of squares.

Sample 1	Sample 2	Sample 3
4	36	81
36	64	121
	100	169
		225
40	200	596	836

3). Sample sum of squares:

where is the total for ith sample.

8	24	48	80
64	576	2304	2944
32	192	576	800

4). Error sum of squares: ESS=TSS-SSS

ESS=124.8889-88.8889

5) Total df=9-1=8

Samles =3-1=2

Error=8-2=6

We shall now construct the ANOVA table:

Source of Variation	SS	df	MS	F	P-value	F crit
Samples	88.88889	2	44.44444	7.407407	0.023952	5.143253
Error	36	6	6
Total	124.8889	8

Since the F-calculated value is greater than the Critical value, we reject the Null hypothesis. Hence, we conclude that there is enough evidence for difference in population means.