Question

2. A least squares adjustment is computed twice on a data set. When the data is minimally constrained with 10 degrees of freedom, a variance of 1.07 is obtained. In the second run, the fully constrained network has 13 degrees of freedom with a variance of 1.56. The a priori

estimate for the reference variances in both adjustments are one; that is,

(1) What is the 95% confidence interval for the reference variance in the minimally constrained adjustment? The population variance is one. Does this interval contain one?

(2) What is the 95% confidence interval for the ratio of the two variances? Is there reason to be concerned about the consistency of the control? Statistically justify your response.

Answer 1

(a) Here we need to find the ratio of the variances and so it is given that

variance of 1st sample = 1.07 with n1= 11 (df = n-1 , so., 10 = n-1 so., n=11)

&

variance of 2nd sample = 1.56 with n2= 14

so., the hypothesis under test is

σ12 = σ22 = 1

And so the F test stat = σ12 / σ22  

= 1.07/1.56 =0.685897

F stat = 0.685897

Now looking at the F table to calculate the Fcritical

The numerator's df = 10 & denominator's df = 13 is given

So., Fcritical = 0.28 & 3.37

While F stat is between Fcritical values and so we can say that we cannot reject Ho

And so we can say that σ12 = σ22 = 1

(b) Now in the minimally constrained condition as well we take the population variance as 1 and check for the critical values of F stat

Since here we have found the Fcritical range from 0.28 to 3.37

and so we can say that the range includes 1 and so we cannot reject the Ho

and so here as well we can say that the population variance is 1

Hope the above answer has helped you in understanding the problem. Please upvote the ans if it has really helped you. Good Luck!!