Question

A random sample of leading companies in South Korea gave the following percentage yields based on assets.

2.1	1.9	4.2	1.2	0.5	3.6	2.4
0.2	1.7	1.8	1.4	5.4	1.1

Use a calculator to verify that s² ≈ 2.200 for these South Korean companies.

Another random sample of leading companies in Sweden gave the following percentage yields based on assets.

2.9

3.7

3.3

1.1

3.5

2.8

2.3

3.5

2.8

Use a calculator to verify that s² ≈ 0.642 for these Swedish companies.

Test the claim that the population variance of percentage yields on assets for South Korean companies is higher than that for companies in Sweden. Use a 5% level of significance. How could your test conclusion relate to an economist's question regarding volatility of corporate productivity of large companies in South Korea compared with those in Sweden?

(a) What is the level of significance?

State the null and alternate hypotheses.

H_o: σ₁² = σ₂²; H₁: σ₁² > σ₂²

H_o: σ₁² > σ₂²; H₁: σ₁² = σ₂²    

H_o: σ₂² = σ₁²; H₁: σ₂² > σ₁²

H_o: σ₁² = σ₂²; H₁: σ₁² ≠ σ₂²

(b) Find the value of the sample F statistic. (Use 2 decimal places.)


What are the degrees of freedom?

dfN
dfD

What assumptions are you making about the original distribution?

The populations follow independent normal distributions.

The populations follow dependent normal distributions. We have random samples from each population.    

The populations follow independent chi-square distributions. We have random samples from each population.

The populations follow independent normal distributions. We have random samples from each population.

(c) Find or estimate the P-value of the sample test statistic.

p-value > 0.100

0.050 < p-value < 0.100    

0.025 < p-value < 0.050

0.010 < p-value < 0.025

0.001 < p-value < 0.010

p-value < 0.001

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?

At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.    

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

(e) Interpret your conclusion in the context of the application.

Fail to reject the null hypothesis, there is sufficient evidence that the variance in percentage yields on assets is greater in the South Korean companies.

Reject the null hypothesis, there is insufficient evidence that the variance in percentage yields on assets is greater in the South Korean companies.    

Reject the null hypothesis, there is sufficient evidence that the variance in percentage yields on assets is greater in the South Korean companies.

Fail to reject the null hypothesis, there is insufficient evidence that the variance in percentage yields on assets is greater in the South Korean companies.

Answer 1

a)

level of significance =0.05

H_o: σ₁² = σ₂²; H₁: σ₁² > σ₂²

b)

value of the sample F statistic =2.2/0.642=3.43

dfN=12

dfD =8

The populations follow independent normal distributions. We have random samples from each population.

c) 0.025 < p-value < 0.050
d)

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.   

e)

Reject the null hypothesis, there is sufficient evidence that the variance in percentage yields on assets is greater in the South Korean companies.