You don't need to be rich to buy a few shares in a mutual fund.The question is, how reliable are mutual funds asinvestments? This depends on the type of fund you buy. Thefollowing data are based on information taken from a mutual fundguide available in most libraries.
A random sample of percentage annual returns for mutual fundsholding stocks in aggressive-growth small companies is shownbelow.
| -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 ≈ 349.621for the sample of aggressive-growth small company funds.
Another random sample of percentage annual returns for mutual fundsholding value (i.e., market underpriced) stocks in large companiesis 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 ≈ 136.311for value stocks in large companies.
Test the claim that the population variance for mutual fundsholding aggressive-growth in small companies is larger than thepopulation variance for mutual funds holding value stocks in largecompanies. Use a 5% level of significance. How could your testconclusion relate to the question of reliability ofreturns for each type of mutual fund?
(a) What is the level of significance?
State the null and alternate hypotheses.
Ho: σ12 =σ22; H1:σ12 >σ22Ho:σ12 >σ22; H1:σ12 =σ22    Ho:σ22 =σ12; H1:σ22 >σ12Ho:σ12 =σ22; H1:σ12 ≠σ22
(b) Find the value of the sample F statistic. (Use 2decimal places.)
What are the degrees of freedom?
What assumptions are you making about the originaldistribution?
The populations follow independent normal distributions.Thepopulations follow dependent normal distributions. We have randomsamples from each population.    Thepopulations follow independent normal distributions. We have randomsamples from each population.The populations follow independentchi-square distributions. We have random samples from eachpopulation.
(c) Find or estimate the P-value of the sample teststatistic. (Use 4 decimal places.)
p-value > 0.1000.050 < p-value <0.100Â Â Â Â 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 orfail to reject the null hypothesis?
At the α = 0.05 level, we reject the null hypothesisand conclude the data are not statistically significant.At theα = 0.05 level, we reject the null hypothesis and concludethe data are statistically significant.    Atthe α = 0.05 level, we fail to reject the null hypothesisand conclude the data are not statistically significant.At theα = 0.05 level, we fail to reject the null hypothesis andconclude the data are statistically significant.
(e) Interpret your conclusion in the context of theapplication.
Fail to reject the null hypothesis, there is sufficient evidencethat the variance in percentage annual returns for mutual funds isgreater in the aggressive-growth in small companies.
Reject the null hypothesis, there is insufficient evidence thatthe variance in percentage annual returns for mutual funds isgreater in the aggressive-growth in smallcompanies.   Â
Reject the null hypothesis, there is sufficient evidence thatthe variance in percentage annual returns for mutual funds isgreater in the aggressive-growth in small companies.
Fail to reject the null hypothesis, there is insufficientevidence that the variance in percentage annual returns for mutualfunds is greater in the aggressive-growth in small companies.
