Question

A portfolio manager claims that the mean annual return on one of the mutual funds he manages equals 8%. To substantiate his claim, he states that over the past 10 years, the mean annual return for the mutual fund has been 9.5% with a sample standard deviation of 1.5%. Assume annual returns are normally distributed.

a. Do a hypothesis test to test the portfolio manager's claim a 5% significance level.

b. Find 95% confidence interval for the population mean annual mutual fund return. Confirm that the conclusion from the hypothesis test you did in part (a) is consistent with the conclusion you would draw from the confidence interval.  

Answer 1

a.

Let X:annual return for the mutual fund.

assuming the normality of X we go as follows

So the hypothesis are

The construction of the test statistic is as follows

The value of the test statistic under H0 is

The p-value of the test is

Clearly 5% level of significance the p-value <0.05.Thus we reject the null hypothesis.That means there is enough evidence to assume that the mean annual return for the mutual fund is significantly differ from the value 8%.

b.

The construction of the 95% CI of is as follows

clearly here also the value 8% does not belong to the interval.So our decision in a. is correct.