Question

A plating company has two silver plating systems with variances σ12 and σ22. You, as the manager, desired to compare the variability in the silver plating done by System-1 with that done by System-2. An independent random sample of size n1= 12 of the System-1 yields s1 = 0.038 mil and sample of size n2= 10 of System-2 yields s2 = 0.042 mil. We need to decide whether σ12= σ22 with α = 0.05. What is the rejection region?

Answer 1

The rejection region is

`F_(cal) < F_{{1-(\alpha)/(2)} , df_1 , df_2 }`

or
`F_(cal) > F_{{(\alpha)/(2)} , df_1 , df_2 }`

and

From the value obtained we see that

`F_(cal) <   F_{(\alpha )/(n) , df_1 , df_2 }` Hence

The decision rule

Fail to reject the null hypothesis

The conclusion

This no sufficient evidence to conclude that there is a difference between the two variance

Explanation:

From the question we are told that

The first sample size is
`n_1 =  12`

The first sample standard deviation is
`s_1 =  0.038 \  mil`

The second sample size is
`n_2 =  10`

The first sample standard deviation is
`s_2 =  0.042 \  mil`

The significance level is
`\alpha =0.05`

The null hypothesis is
`H_o : \sigma^2_1  = \sigma^2_2`

The alternative hypothesis is
`H_o : \sigma^2_1 \\e  \sigma^2_2`

Generally the test statistics is mathematically represented as

`F_(cal) =  (s_1^2 )/(s_2^2)`

=>
`F_(cal) =  ( 0.038^2 )/(0.042^2)`

=>
`F_(cal) =  0.81859`

Generally the first degree of freedom is
`df_1 =  n_1 -1  =  12-1 = 11`

Generally the second degree of freedom is
`df_2 =  n_2 -1  =  10 -1 = 9`

From the F-table the critical value of
`(\alpha)/(2)` at the degrees of freedom of
`df_1 = 11` and
`df_2 =9` is

`F_{(\alpha )/(n) , df_1 , df_2 } =  3.91207`

The rejection region is

`F_(cal) < F_{{1-(\alpha)/(2)} , df_1 , df_2 }`

or
`F_(cal) > F_{{(\alpha)/(2)} , df_1 , df_2 }`

and

From the value obtained we see that

`F_(cal) <   F_{(\alpha )/(n) , df_1 , df_2 }` Hence

The decision rule

Fail to reject the null hypothesis

The conclusion

This no sufficient evidence to conclude that there is a difference between the two variance