UNC-Wilmington
Department of Economics and Finance
ECN 377
Dr. Chris Dumas
Homework 8 Solutions
Multiple Choice
1) b, Chi-square test
2) b, Chi-square test
3) d, F-test
Problem Section
4) In investment analysis, the variance in the return on an investment is a critical piece of information. Suppose that the
variance in the return on investment A has been 2.34 in the past, but suppose a recent event in the market may have affected
the return on investment A. Based on a sample size of 41 randomly sampled observations on this investment, the sample
variance in its return now appears to be 2.39. Use the appropriate hypothesis test to determine whether there has been a
significant increase in the variance of the return for investment A. Assume a Confidence Level of 95%. SHOW YOUR
WORK, including the null hypothesis and the alternative hypothesis that you are testing!
σ2past = 2.34 (given value)
n = 41
s2 = 2.39
H0: σ2now = (σ2past = 2.34)
H1: σ2now > (σ2past = 2.34) ===> one-sided test
𝑠2
2
𝜒𝑡𝑒𝑠𝑡
= (𝑛 − 1) ∙ 𝜎2
= (41 – 1)*(2.39/2.34) = 40.85
d.f. = n – 1 = 40
Confidence Level of 95% ==> α = 0.05
2
2
find χ critical from the χ -table based on d.f. = 40 and the Significance Level α = 0.05
χ2critical = 55.76
χ 2test < χ2critical ===> DO NOT Reject H0
Conclude: There has NOT been a significant increase in the variance.
5) Suppose you also have a sample of 21 randomly sampled observations on investment B, and the sample variance in its
return is currently 3.78 . Based on your results from the problem above, conduct the appropriate hypothesis test to determine
whether the variance in investment B is currently significantly larger than the variance in investment A. Assume a
Confidence Level of 95%. SHOW YOUR WORK, including the null hypothesis and the alternative hypothesis that you are
testing!
σ2A = population A variance (unknown)
nA = 41
s2A = 2.39
σ2B = population B variance (unknown)
nB = 21
s2B = 3.78
H0: σ2B = σ2A
H1: σ2B > σ2A===> one-sided
𝑠2
𝐹𝑡𝑒𝑠𝑡 = 𝑠𝐵2 = 3.78/2.39 = 1.58
𝐴
d.f.numerator = 21 – 1= 20 and d.f.denominator = 41 – 1=40
Confidence Level of 95% ==> α = 0.05
find Fcritical from the F-table based on d.f.numerator, d.f.denominator, and the Significance Level “α”
Fcritical = 1.84
If Ftest < Fcritical ===> DO NOT Reject H0.
Conclude: The variance of investment B is NOT significantly larger than the variance of investment A.
1
UNC-Wilmington
Department of Economics and Finance
ECN 377
Dr. Chris Dumas
6) Suppose your consulting firm is hired to determine whether there is more income inequality in the southern United States
relative to the northern United States. To assess this, you collect data on mean household income for a random sample of 260
counties in the northern United States and a random sample of 140 counties in the southern U.S. The sample variance in
mean household income for the northern counties is 1520, and the sample variance for the southern states is 2467. Use the
appropriate hypothesis test to determine whether there is a statistically significant difference in the variance of household
income between northern and southern states. Assume a Confidence Level of 95%. SHOW YOUR WORK, including the
null hypothesis and the alternative hypothesis that you are testing!
σ2N = North population variance (unknown)
nN = 260
s2N = 1520
σ2S = South population variance (unknown)
nS = 140
s2S = 2467
H0: σ2S = σ2N
H1: σ2S > σ2N ===> one-sided
𝑠2
𝐹𝑡𝑒𝑠𝑡 = 𝑠𝑆2 = 2467/1520 = 1.62
𝑁
d.f.numerator = 140 – 1= 139 and d.f.denominator = 260 – 1 = 259
Confidence Level of 95% ==> α = 0.05
find Fcritical from the F-table based on d.f.numerator, d.f.denominator, and the Significance Level “α”
Fcritical = 1.39
If Ftest > Fcritical ===> Reject H0.
Conclude: The variance of household income in Southern states is significantly larger than the variance
of household income in Northern states.
7) You work for a large auto company that sells autos in Europe and China. Your boss wants to know whether auto prices
are more variable in Chinese cities relative to European cities. You have good data on your auto sales in European cities, and
you know that the variance in auto prices in those cities is 4500. Your data from China is not as good. You have data on
auto prices in a sample of 13 Chinese cities. In these cities, the sample variance of auto prices is 6300. Use the appropriate
hypothesis test to determine whether the variance in auto prices in Chinese cities is significantly larger than the variance in
European cities. Assume a Confidence Level of 95%. SHOW YOUR WORK, including the null hypothesis and the
alternative hypothesis that you are testing!
σ2Europe = 4500 (given value)
NChina = 13
s2China = 6300
H0: σ2China = (σ2Europe = 4500)
H1: σ2China > (σ2 Europe = 4500) ===> one-sided test
𝑠2
2
𝜒𝑡𝑒𝑠𝑡
= (𝑛 − 1) ∙ 𝜎2
= (13 – 1)*(6300/4500) = 16.8
d.f. = n – 1 = 12
Confidence Level of 95% ==> α = 0.05
2
2
find χ critical from the χ -table based on d.f. = 12 and the Significance Level α = 0.05
χ2critical = 21.03
χ 2test < χ2critical ===> DO NOT Reject H0
Conclusion: The variance in auto prices in Chinese cities is NOT significantly larger than the variance in
auto prices in European cities.
2