Question

You don't need to be rich to buy a few shares in a mutual fund. The question is, how reliable are mutual funds as investments? This depends on the type of fund you buy. The following data are based on information taken from a mutual fund guide available in most libraries.

A random sample of percentage annual returns for mutual funds holding stocks in aggressive-growth small companies is shown below.

-1.6	14.6	41.7	17.3	-16.9	4.4	32.6	-7.3	16.2	2.8	34.3
-10.6	8.4	-7.0	-2.3	-18.5	25.0	-9.8	-7.8	-24.6	22.8

Use a calculator to verify that s² ≈ 349.621 for the sample of aggressive-growth small company funds.

Another random sample of percentage annual returns for mutual funds holding value (i.e., market underpriced) stocks in large companies is shown below.

16.5	0.4	7.1	-1.1	-3.2	19.4	-2.5	15.9	32.6	22.1	3.4
-0.5	-8.3	25.8	-4.1	14.6	6.5	18.0	21.0	0.2	-1.6

Use a calculator to verify that s² ≈ 136.311 for value stocks in large companies.

Test the claim that the population variance for mutual funds holding aggressive-growth in small companies is larger than the population variance for mutual funds holding value stocks in large companies. Use a 5% level of significance. How could your test conclusion relate to the question of reliability of returns for each type of mutual fund?

(a) What is the level of significance?

State the null and alternate hypotheses.

H_o: σ₁² = σ₂²; H₁: σ₁² > σ₂²H_o: σ₁² > σ₂²; H₁: σ₁² = σ₂²    H_o: σ₂² = σ₁²; H₁: σ₂² > σ₁²H_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 normal distributions. We have random samples from each population.The populations follow independent chi-square distributions. We have random samples from each population.

(c) Find or estimate the P-value of the sample test statistic. (Use 4 decimal places.)

p-value > 0.1000.050 < p-value < 0.100    0.025 < p-value < 0.0500.010 < p-value < 0.0250.001 < p-value < 0.010p-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 annual returns for mutual funds is greater in the aggressive-growth in small companies.

Reject the null hypothesis, there is insufficient evidence that the variance in percentage annual returns for mutual funds is greater in the aggressive-growth in small companies.    

Reject the null hypothesis, there is sufficient evidence that the variance in percentage annual returns for mutual funds is greater in the aggressive-growth in small companies.

Fail to reject the null hypothesis, there is insufficient evidence that the variance in percentage annual returns for mutual funds is greater in the aggressive-growth in small companies.

Answer 1

The statistical software output for this problem is:

Two sample variance summary hypothesis test:
σ₁² : Variance of population 1
σ₂² : Variance of population 2
σ₁²/σ₂² : Ratio of two variances
H₀ : σ₁²/σ₂² = 1
H_A : σ₁²/σ₂² > 1

Hypothesis test results:

Ratio	Num. DF	Den. DF	Sample Ratio	F-Stat	P-value
σ12/σ22	20	20	2.5648774	2.5648774	0.0205

Hence,

a) Level of significance = 0.05

Hypotheses: Ho: σ12 = σ22; H1: σ12 > σ22

b) F statistic = 2.56

DFN = 20

DFD = 20

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

c) 0.010 < p-value < 0.025

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 annual returns for mutual funds is greater in the aggressive-growth in small companies.