Question

In a test of two population means - μ1μ1 versus μ2μ2 - with unknown variances σ21σ12 and σ22σ22, two independent samples of n1=8n1=8 and n2=10n2=10 were taken. The data is given below. Both populations are normally distributed.

Sample From Population 1: 11, 7, 14, 14, 19, 16, 16, 16 ;
Sample From Population 2: 16, 15, 19, 16, 16, 14, 19, 20, 20, 18

You wish to test the hypothesis that both populations have the same variance. Choose the correct statistical hypotheses.
A. H0:σ21≠σ22HA:σ21=σ22H0:σ12≠σ22HA:σ12=σ22
B. H0:σ21=σ22HA:σ21≠σ22H0:σ12=σ22HA:σ12≠σ22
C. H0:σ21=σ22HA:σ21<σ22H0:σ12=σ22HA:σ12<σ22
D. H0:S21=S22HA:S21≠S22H0:S12=S22HA:S12≠S22
E. H0:σ21=σ22HA:σ21>σ22H0:σ12=σ22HA:σ12>σ22

(b) Determine the value of the test statistic for this test. Use at least two decimals in your answer.
Test Statistic =

(c) Determine the P-value for this test, to at least three decimal places.

Answer 1

Part a

The null and alternative hypotheses for this test are given as below:

B. H₀: σ₁² = σ₂² versus H_A: σ₁² ≠ σ₂²

[...because claim is given as the both populations have same variance.]

Part b

The test statistic value is given as below:

F = S1^2/S2^2

From given data, we have

S1^2 = 13.55357143

S2^2 = 4.677777778

F = 13.55357143 / 4.677777778

F = 2.89743807

Test statistic = F = 2.89743807

Part c

We have

F = 2.89743807

n1 = 8

n2 = 10

df1 = n1 – 1 = 8 – 1 = 7

df2 = n2 – 1 = 10 – 1 = 9

So, P-value by using F-table or excel is given as below:

P-value = 0.1400