Question

Eurelia

Sample size = 11, Mean Rent = $453, Standard Deviation = $56

Begonia

Sample size = 8, Meant Rent = $380, Standard Deviation = $84

a) Using the F tables, test whether the variances are too different to use the pooled variance method rather than Satterthwaite's method

b) Test the 5% level whether the mean rent in Eurelia is significantly higher than the mean rent in Begonia

c) Estimate the standard error of the difference in mean rents

Answer 1

(a)

Data:       

n1 = 11      

n2 = 8      

s1^2 = 3136      

s2^2 = 7056      

Hypotheses:      

Ho: σ1^2 = σ2^2      

Ha: σ1^2 ≠ σ2^2      

Decision Rule:      

α = 0.05      

Numerator DOF = 11 - 1 = 10    

Denominator DOF = 8 - 1 = 7    

Lower Critical F- score = 0.2532    

Upper Critical F- score = 4.7611    

Reject Ho if F < 0.253176 or F > 4.7611   

Test Statistic:      

F = s1^2 / s2^2 = 3136/7056 =   0.4444   

p- value = 0.881588      

Decision (in terms of the hypotheses):    

Since 0.253176 < 0.4444 < 4.7611 we fail to reject Ho

Conclusion (in terms of the problem):    

There is no sufficient evidence that the population variances are significantly different.

(b)

Data:     

n1 = 11    

n2 = 8    

x1-bar = 453    

x2-bar = 380    

s1 = 56    

s2 = 84    

Hypotheses:    

Ho: μ1 ≤ μ2    

Ha: μ1 > μ2    

Decision Rule:    

α = 0.05    

Degrees of freedom = 11 + 8 - 2 = 17

Critical t- score = 1.73960672   

Reject Ho if t > 1.73960672   

Test Statistic:    

Pooled SD, s = √[{(n1 - 1) s1^2 + (n2 - 1) s2^2} / (n1 + n2 - 2)] = √(((11 - 1) * 56^2 + (8 - 1) * 84^2)/(11 + 8 - 2)) = 68.92109726

SE = s * √{(1 /n1) + (1 /n2)} = 68.9210972566371 * √((1/11) + (1/8)) = 32.02489005

t = (x1-bar -x2-bar)/SE = (453 - 380)/32.0248900527028 = 2.279476991

p- value = 0.01791149    

Decision (in terms of the hypotheses):

Since 2.27947699 > 1.739606716 we reject Ho and accept Ha

Conclusion (in terms of the problem):

There is sufficient evidence that the mean rent in Eurelia is significantly higher than the mean rent in Begonia.

(c)

Pooled SD, s = √[{(n1 - 1) s1^2 + (n2 - 1) s2^2} / (n1 + n2 - 2)] = √(((11 - 1) * 56^2 + (8 - 1) * 84^2)/(11 + 8 - 2)) = 68.92109726

SE = s * √{(1 /n1) + (1 /n2)} = 68.9210972566371 * √((1/11) + (1/8)) = 32.02489005