Question

The data below are from an independent-measures experiment comparing three different treatment conditions.

Treatment 1     Treatment 2     Treatment 3

0                      1                      4

0                      4                      3          G=24

0                      1                      6          SX squared=92

2                      0                      3

                                        _________________________________

                                                T=2                 T=6                 T=16

                                                SS=3                SS=9                SS=6

1. Use an ANOVA with a=.05 to determine whether these data indicate any significant differences among treatments.

2. Use the Tukey’s HSD to determine which of the treatments are significantly

different from each other at the .05 level.

PLEASE SHOW STEP BY STEP ANSWER

Answer 1

a:

Total number of observations:

N = 4+4+4 = 12

The grand mean is

Let $ar{x_{i}}$ shows the mean of the ith group. So

$ar{x}_{1}=rac{2}{4}=0.5,ar{x}_{2}=rac{6}{4}=1.5,ar{x}_{3}=rac{16}{4}=4$

Now,

betueen-721 (TI-T

and

Therefore

$SS_{total}=SS_{between}+SS_{within}=44$

----

Since there are 3 different groups so we have k=3. Therefore degree of freedoms are:

$df_{between}=k-1=3-1=2$

dJwithin

$df_{total}=9+2=11$

-------------

Now

$MS_{between}=rac{SS_{between}}{df_{between}}=13$

$MS_{within}=rac{SS_{within}}{df_{within}}=2$

F test statistics is

$F=rac{MS_{between}}{MS_{within}}=6.5$

So p-value of the test is 0.0179. Since P-value is less than 0.05 so we reject the null hypothesis. That is on the basis of sample evidence we can conclude that populations are different.

(b)

Critical value for $\alpha=0.05$, df=9 and k=3 is

$q=3.95$

So Tukey's HSD will be

Following table shows which difference is significant:

xbarj groups (i-j)xbari 0.5 0.5 1.5 mu1-mu2 mu1-mu3 mu2-mu3 ni 4 4 4 nj 4 4 4 HSD xbari-xbarj Lower limit Upper limit Significant(Yes/No) 2.79 2.79 2.79 1 3.5 -2.5 3.79 6.29 -5.29 1.79 4 4 No Yes No 0.29