Answer to Question #136590 in Statistics and Probability for cccc

Question

The following data show the effect of 4 operators, chosen randomly, on the output of a particular machine.
operator 1: 175.4 171.7 173.0 170.5
operator 2: 168.5 162.7 165.0 164.1
operator 3: 170.1 173.4 175.7 170.7
operator 4: 175.2 175.7 180.1 183.7
(a) Perform al analysis of variance at the 0.01 level of significance. (b) Compute an estimate of the operator variance component and the experimental error variance component. What inference(s) can you make based on the computed variance components?

Answer 1

(a)

We are given a table with 4 columns. It shows us the effect of 4 operators,

chosen randomly, on the output of a particular machine. There are 4 sampe

data in the table for each operatot, so we can see that k = 4 and n = 4. We

want to use the ANOVA, so we have to find the means of operators and the

overall mean, The results are:

$\mu_1$ = 172.65

$\mu_2$ = 165.075

$\mu_3$ = 172.475

$\mu_4$ = 178.675

The overall mean is $\mu$ = 172.219. We want to test these hypothesis:

H₀: $\sigma_a^2$ =0

H₁:$\sigma_a^2$ ≠0

In order to perform the ANOVA, we have to find the values of SST,SSA and

SSE and we have to determine their degrees of freedom. First sum we are

going to find is SST:

SST =$\sum^4_{i=1}\sum^4_{i=1}(y_{ij}-\mu)^2$= 471.6644

SST has kn —1 degrees of freedom. Since k= n =4, we get that SST has

15 degrees of freedom.

The next sum whose value we are going to determine is SSA:

SSA=$n*\sum^4_{i=1}=4\sum^4_{i=1}(\mu_{i}-\mu)^2=371.8719$

SSA has k—1 degrees of freedom. Since k = 4,we get that SSA has 3

degrees of freedom.

The only sum left is SSE. Its value is:

SSE = SST — SSA= 99.7925

SSE has k(n — 1) degrees of freedom. Since k =n=4, we get that SSE has 12 degrees of freedom.

Now we have everything we need for creating ANOVA table. The table is

given by:

Source | Sum Sq |Deg| Mean Sq |F

Treatment | 371.8719 | 3 | 123.9573| 14.879

Error__ | 99.7925 |12 | 8.331

Total |471.6644 | 15 |

We get that the value of F statistic is 14.879. F- statistic has 3 and 12

degrees of freedom. Since we want to determine p-value, we are going to use

F-distribution applet. The result we get is

p-value is 0.00024. since the given significance level Is 0.05, we can see

that p < 0.05. This is why we reject null hypothesis. There is evidence to

coneldue that the operators are different

(b)

We have to find an estimate of the operator variance component and the

experimental error variance component. Since we already created the ANOVA

table, we just have to read the values and use the appropriate formula from

chapter 13. As a result we get:

$\sigma^2$ = MSE =8.331

$\sigma_a^2$ =(MSA- MSE)/4=(123.9573 — 8.331)/4=28.9066