Question

Given the null and alternative hypotheses and sample information shown​ below, complete parts a and b.

Given the null and alternative hypotheses and sample information shown below, complete parts a and b. Ho: 07 = 63 HA: 0; #o2 Sample 1 n = 12 Sy = 13 Sample 2 n2 = 21 Sy = 34 a. If a=0.05, state the decision rule for the hypothesis. If the calculated test statistic F is more extreme than 3.23, reject Ho, otherwise do not reject Ho- (Round to two decimal places as needed. Use a comma to separate answers as needed.) b. Test the hypothesis and indicate whether the null hypothesis should be rejected. Determine the test statistic. F = 8.03 (Round to two decimal places as needed.) What is the conclusion at the significance level of 0.05? O A. Do not reject the null hypothesis. There is sufficient evidence to conclude that the population variances are different. O B. Reject the null hypothesis. There is sufficient evidence to conclude that the population variances are different. O C. Do not reject the null hypothesis. There is insufficient evidence to conclude that the population variances are different. OD. Reject the null hypothesis. There is insufficient evidence to conclude that the population variances are different.

Can you please check the answers and show the steps?

Answer 1

Since , Ha = $\sigma$ 12  $\neq$$\sigma$22

It is a two tailed test.
For a two tailed test in F Statistic , there are two cases:
1. fcal  $\geq$ F$\alpha /2,$ n1-1, n2-1 or   fcal  $\leq$ F1 - $\alpha /2,$ n1-1, n2-1
s12 / s22   $\geq$ F$\alpha /2,$ n1-1, n2-1   or  s12 / s22   $\leq$ F1 - $\alpha /2,$ n1-1, n2-1
Decision Rule: Reject H0

2. F1 - $\alpha /2,$ n1-1, n2-1 < fcal <F$\alpha /2,$ n1-1, n2-1  
F1 - $\alpha /2,$ n1-1, n2-1 < s12 / s22 <F$\alpha /2,$ n1-1, n2-1  
Decision Rule: Do Not Reject H0

(Note : We have taken $\alpha /2,$ because it is a two tail test.)
Here , n1-1 is the degree of freedom in sample 1
and n2-1 is the degree of freedom in sample 2

(a) ........F is more than 3.37 , reject H0
Reason : At $\alpha$ = 0.05
F$\alpha /2,$ n1-1, n2-1  = F0.05/2, 12-1, 21-1 (given n1 =12 , n2 = 21 )
= F0.025, 11, 20
Using Table to look at this value,  
Since v1 = 11 is not provided in table , so we will look at the closerer and smallest value which is v1 = 10.
Again for 0.025 , there is no value so we will look at 0.01 which is nearer and smallest
Thus , for F0.025, 11, 20 , we will look at F0.01, 10, 20 which is 3.37
Hence, using fcal  $\geq$ F$\alpha /2,$ n1-1, n2-1
fcal  $\geq$ 3.37 to reject null hypothesis (H0)

(b) F = 0.15
Reason: F = fcal =s12 / s22
= (13)2 / (34)2
= 0.146 or 0.15 ( round to two decimals)
Conclusion : Part C Do not Reject the null hypothesis . There is insufficient evidence to conclude that population variances are different.
Reason : As in this part we have not calculated (F1 - $\alpha /2,$ n1-1, n2-1  ) value yet to decide whether to reject or not.