Question

Trading skills of institutional investors. The trading skills of institutional stock investors were quantified and analyzed in The Journal of Finance (April 2011). The study focused on “round-trip” trades, i.e., trades in which the same stock was both bought and sold in the same quarter. Consider a random sample of 200 round-trip trades made by institutional investors. Suppose the sample mean rate of return is 2.95% and the sample standard deviation is 8.82%. If the true mean rate of return of round-trip trades is positive, then the population of institutional investors is considered to have performed successfully.

a. Specify the null and alternative hypotheses for determining whether the population of institutional investors performed successfully.

b. Find the rejection region for the test using\(\alpha = 0.05\).

c. Interpret the value of\(\alpha \)in the words of the problem.

d. A Minitab printout of the analysis is shown below. Locate the test statistic and p-value on the printout. (Note: For large samples, z ≈ t.)

e. Give the appropriate conclusion in the words of the problem.

Short answer

a.\({H_0}:\mu = 0\)and\({H_a}:\mu > 0\)are the null and the alternative hypothesis.

b. The rejection region is \({z_c} > 1.645\).

c. The probability of the true mean rate of return of round-trip trades is positive.

d.From the MINITAB output, the test statistic is 4.73, and the p-value is 0.000.

e. It can be concluded that the population of institutional investors did not perform successfully at the significance level \(\alpha = 0.05\).

Step-by-step solution

Given information

The sample size is 200, the sample mean is 2.95, and the sample standard deviation is 8.82.

Also, MINITAB output is as follows

Setting up the null and alternative hypothesis

a.

Null hypothesis:

\({H_0}:\mu = 0\)

That is, the population of institutional investors performed successfully.

Alternative hypothesis:

\({H_a}:\mu > 0\)

That is, the population of institutional investors did not perform successfully.

Finding the rejection region

b.

Here, the test is the right tail, and the significance level is 0.05. The critical value \({z_{0.05}}\) is obtained from the standard normal table.

Thus, the required \({z_{0.05}}\) critical value is 1.645.

The rejection region for the right tail test is\({z_c} > {z_a}\)

Hence, the rejection region is \({z_c} > 1.645\).

Interpreting the value of \(\alpha \) \(\)

c.

The probability of Type I error is denoted as\(\alpha \). In other words, it is the probability of the error committed to rejecting a null hypothesis\(\left( {{H_0}} \right)\)when it is true.

In the study, the interpretation \(\alpha \) is that the probability of the population of institutional investors did not perform successfully. That is, the probability of the true mean rate of return of round-trip trades is positive.

Finding the test statistic and the p-value

d.

From the MINITAB output, the test statistic is 4.73, and the p-value is 0.000.

Conclusion

e.

Here, the p-value is 0.000, which is less than the value \(\alpha = 0.05\).

If the p-value <\(\alpha \), then the null hypothesis is rejected.

Hence, Reject the null hypothesis\({H_0}\)

Thus, it can be concluded that the population of institutional investors did not perform successfully at the significance level\(\alpha = 0.05\).

That is, the true mean rate of return of round-trip trades is positive. \(\)