Suppose a mutual fund qualifies as having moderate risk if the standard deviation of its monthly rate of return is less than 3%. a mutual-fund rating agency randomly selects 27 months and determines the rate of return for a certain fund. the standard deviation of the rate of return is computed to be 2.19%. is there sufficient evidence to conclude that the fund has moderate risk at the alpha equals 0.05 level of significance? a normal probability plot indicates that the monthly rates of return are normally distributed.

There is sufficient evidence to conclude that the fund has moderate risk at the 0.05 level of significance

Explanation:

Given:

Sample size = 27months

n = 27

Let sigma = population standard deviation

Sigma = 3% = 0.03

s = sample standard deviation

s =2.19% = 0.0219

Alpha = 0.05

The mutual fund manager claims that his fund has moderate risk.

Hypothesis:

H0: Population standard deviation is equal to sample standard deviation.

H0 = sigma = 0.03

H1: Population standard deviation is not equal to sample standard deviation.

Alternative hypothesis:

H0< sigma = H0 <0.03

Degree of freedom = n-1 = 27-1 = 26

We would use the chi-square test to check if the standard deviation of a population is equal to a particular value

The test statistic:

x^2 = [(n-1)s^2]/ sigma^2

x^2 = [(27-1)*0.0219^2]/0.03^2

x^2 = [(26)*0.00047961]/0.03^2

x^2 = (0.01246986/0.03^2)

x^2 = 0.01246986/ 0.0009

x^2 = 13.8554

You can compare the value of your x^2 to a (degrees of freedom, alpha) chi squared critical value from a table (you can find such table online) or use an EXCEL formula

Using EXCEL formula to find p-value:

CHISQ.DIST(13.8554, 26, TRUE)

Type in the value of chi square as x,

Degree of freedom as = 26 and

TRUE (cumulative distribution function)

p-value = 0.025143

The level of significance = Alpha = 0.05

The decision rule states:

If p-value is less than or equal to level of significance, reject the null hypothesis.

Otherwise, do not reject the null hypothesis.

Since p-value is less than level of significance, we would reject the null hypothesis.

0.025143 < 0.05

Conclusion:

There is sufficient evidence to conclude that the fund has moderate risk at the 0.05 level of significance.

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