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.
-1.4 | 14.5 | 41.4 | 17.9 | -16.3 | 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 s^{2} ≈ 347.715
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 | 0.8 | 7.2 | -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 s^{2} ≈ 135.972
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}: σ_{1}^{2} = σ_{2}^{2}; H_{1}: σ_{1}^{2} > σ_{2}^{2}
H_{o}: σ_{1}^{2} > σ_{2}^{2}; H_{1}: σ_{1}^{2} = σ_{2}^{2}
H_{o}: σ_{2}^{2} = σ_{1}^{2}; H_{1}: σ_{2}^{2} > σ_{1}^{2}
H_{o}: σ_{1}^{2} = σ_{2}^{2}; H_{1}: σ_{1}^{2} ≠ σ_{2}^{2}
(b) Find the value of the sample F statistic. (Use 2
decimal places.)
What are the degrees of freedom?
df_{N} | |
df_{D} |
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. (Use 4 decimal places.)
p-value > 0.100
0.050 < p-value < 0.100
0.025 < p-value < 0.0500
.010 < p-value < 0.0250
.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 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.
a)Ho: σ1^{2} = σ2^{2}; H1: σ1^{2} > σ2^{2}
b)
value of the sample F statistic =(s1/s2)^{2} =(347.715/135.972)=2.56
dfN =21-1=20
dfD =21-1=20
The populations follow independent normal distributions. We have random samples from each population.
c)
.010 < p-value < 0.0250
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.
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