Question

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.

Answer 1

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

Step-by-step 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.