Question

Countries	Continent	GDP(log)		Countries	Continents	GDP(log)
Austria	Europe	10.70044938		Afghanistan	Asia	7.541026821
Belgium	Europe	10.61170212		Armenia	Asia	8.926230102
Bulgaria	Europe	9.661098603		Bangladesh	Asia	7.956283275
Croatia	Europe	9.906639101		Cambodia	Asia	7.987363009
Cyprus	Europe	10.21807709		China	Asia	9.352236341
Czech Republic	Europe	10.23849978		India	Asia	8.563698921
Denmark	Europe	10.64519906		Indonesia	Asia	9.13285679
Estonia	Europe	10.13189828		Iran (Islamic Republic of)	Asia	9.621790681
Finland	Europe	10.56735151		Israel	Asia	10.33937999
France	Europe	10.52283869		Japan	Asia	10.4805028
Germany	Europe	10.67374994		Jordan	Asia	9.341939048
Greece	Europe	10.10803962		Korea (Republic of)	Asia	10.39537826
Hungary	Europe	10.03949973		Kyrgyzstan	Asia	8.042335304
Iceland	Europe	10.6273945		Lao People's Democratic Republic	Asia	8.448344608
Ireland	Europe	10.71288419		Mongolia	Asia	9.119573483
Italy	Europe	10.43902256		Nepal	Asia	7.683767515
Latvia	Europe	9.99079159		Pakistan	Asia	8.401574648
Lithuania	Europe	10.10571801		Russian Federation	Asia	10.06748423
Luxembourg	Europe	11.38210373		Sri Lanka	Asia	9.151199246
Malta	Europe	10.26909988		Thailand	Asia	9.541930099
Moldova (Republic of)	Europe	8.416385302
Netherlands	Europe	10.71318574

Do you expect a group difference for the mean logarithm of GDP per capita in this comparisons(European countries vs Asian countries)? Why or why not?

Apply an appropriate test to assess whether the mean logarithm of GDP per capita is different between the two groups(Europe and Asia)

Answer 1

Solution

[NOTE: Answers are given below. Detailed Working and Back-up Theory follow at the end.]

Test is carried out at significance level of 5% (i.e., α = 0.05 ).

Conclusion:

There is sufficient evidence to conclude that mean logarithm of GDP per capita is different between the two groups(Europe and Asia). Answer

Summary Statistics

tcal =	5.610313
α =	0.05
tcrit =	2.021075
p-value =	1.67E-06

Back-up Theory and Detailed Working

Let X = GDP(log) for European countries and

      Y = GDP(log) for Asian countries

Then, X ~ N(µ₁, σ₁²) and Y ~ N(µ₂, σ₂²), where σ₁² = σ₂² = σ², say and σ² is unknown.

Claim: Mean logarithm of GDP per capita is different between the two groups(Europe and Asia)

Hypotheses:

Null: H₀: µ1 = µ2 Vs Alternative: H_A: µ1 ≠ µ2 [claim]

Test Statistic:

t = (Xbar - Ybar)/{s√(2/n)} where

s² = (s₁² + s₂²)/2;

Xbar and Ybar are sample averages and s₁,s₂ are sample standard deviations based on n observations each on X and Y.

Calculations

Summary of Excel calculations is given below:

n1 =	22
n2 =	20
Xbar =	10.30371
Ybar =	9.004745
s1 =	0.563347
s2 =	0.911889
s =	0.749398
tcal =	5.610313
α =	0.05
tcrit =	2.021075
p-value =	1.67E-06

Distribution, significance Level α, Critical Value and p-value:

Under H₀, t ~ t_{2n - 2}. Hence, for level of significance α%, Critical Value = upper (α/2)% point of t_{2n - 2} and p-value = P(t_{2n - 2} > | tcal |). α is taken to be 5% (i.e., 0.05).

Using Excel Functions, the above are found to be as shown in the above table.

Decision:

Since | tcal | > tcrit, or equivalently, since p-value < α, H₀ is rejected.

Conclusion:

There is sufficient evidence to suggest that the claim is valid.

DONE