Question

The following observations are from two independent random samples, drawn from normally distributed populations.

Sample 1	[66.73, 66.8, 75.06, 58.09, 54.64, 52.83]
Sample 2	[66.71, 68.17, 66.22, 66.8, 68.81]

Test the null hypothesis H0:σ21=σ22 against the alternative hypothesis HA:σ21≠σ22.

a) Using the larger sample variance in the numerator, calculate the F test statistic.

Round your response to at least 3 decimal places.

   

b) The p-value falls within which one of the following ranges:

		p-value > 0.50
		0.10 < p-value < 0.50
		p-value < 0.10

c) What conclusion can be made at the 10% level of significance?

		There is sufficient evidence to reject the null hypothesis at the 10% level of significance, and therefore evidence that the population variances are not equal to each other.
		There is insufficicent evidence to reject the null hypothesis at the 10% level of significance, and therefore no significant evidence that the population variances are not equal to each other.

Answer 1

Solution-:

By using R-software:

> #Hypothesis:
> #Ho:sigma21=sigma22
> #V/s
> #H1:sigma21 not equal sigma22
> x=c(66.73,66.8,75.06,58.09,54.64,52.83);x
[1] 66.73 66.80 75.06 58.09 54.64 52.83
> y=c(66.71,68.17,66.22,66.8,68.81);y
[1] 66.71 68.17 66.22 66.80 68.81
> var.test(x,y,conf.level=0.90)

F test to compare two variances

data: x and y
F = 61.553, num df = 5, denom df = 4, p-value = 0.001422
alternative hypothesis: true ratio of variances is not equal to 1
90 percent confidence interval:
9.838889 319.591677
sample estimates:
ratio of variances
61.55265

> #Decision Rule: If P-value <0.10(alpha),Reject Ho,at Los.
> #Interpretation: Here, p-value=0.001422< 0.10 (Alpha),so we reject Ho.
> #Conclusion:There is sufficient evidence to reject the null hypothesis at the 10% level of significance, and therefore evidence that the population variances are not equal to each other.

From this output we get,

(a) Test statistic =F = 61.553

(b) The p-value falls within p-value < 0.10

Option (3) is correct.

(c) Conlusion-Here, p-value=0.001422< 0.10 ,so we reject Ho.

There is sufficient evidence to reject the null hypothesis at the 10% level of significance, and therefore evidence that the population variances are not equal to each other.