Question

1. Suppose we have the following information on GMAT scores for business and non-business majors:

Business Majors                     Non-Business Majors

n₁ = 8                                      n₂ = 5

_                                              _

X₁ = 545                                  X₂ = 525

s₁ = 120                                   s₂ = 60

1. Using a 0.05 level of significance, test to see whether the population variances are equal. (4 points)

b. Using a 0.05 level of significance, test the clam that average GMAT scores for business majors is above the average GMAT scores for non-business majors in the population. Assume unequal population variances.

Answer 1

Part a

Here, we have to use the F test for the population variances.

H₀: σ₁² = σ₂² versus H_a: σ₁² ≠ σ₂²

(Two tailed test)

We are given

α = 0.05

S1 = 120

S2 = 60

n1 = 8

n2 = 5

df1 = n1 – 1 = 8 – 1 = 7

df2 = n2 – 1 = 5 – 1 = 4

Test statistic is given as below:

F = S1^2/S2^2 = 120^2/60^2 = 4

P-value = 0.1984

(by using F-table or excel)

P-value > α = 0.05

So, we do not reject the null hypothesis

There is sufficient evidence to conclude that the population variances are equal.

Part b

Solution:

Here, we have to use two sample t test for the difference in population means by assuming unequal population variances.

H₀: µ₁ = µ₂ versus H_a: µ₁ > µ₂

t = (X1bar – X2bar) / sqrt[(S₁² / n₁)+(S₂² / n₂)]

Degrees of freedom = [(S₁²/n₁) + (S₂²/n₂)]^2 / [((S₁²/n₁)^2/(n₁ – 1)) + ((S₂²/n₂)^2/(n₂ – 1))]

WE are given

We are given

α = 0.05

X1bar = 545

X2bar = 525

S1 = 120

S2 = 60

n1 = 8

n2 = 5

Degrees of freedom = [(120^2/8) + (60^2/5)]^2 / [((120^2/8)^2/(8 – 1)) + ((60^2/5)^2/(5 – 1))]

Degrees of freedom = 10.7188

Degrees of freedom = 10

t = (X1bar – X2bar) / sqrt[(S₁² / n₁)+(S₂² / n₂)]

t = (545 – 525) / sqrt[(120^2 / 8)+(60^2 / 5)]

t = 20/50.1996

t = 0.3984

P-value = 0.3494

(by using t-table)

P-value > α = 0.05

So, we do not reject the null hypothesis

There is insufficient evidence to conclude that that average GMAT scores for business majors is above the average GMAT scores for non-business majors in the population.