Question

1. Over a period of 100 randomly chosen trading days in 1993, a basket of small growth stocks returned an average of 13.37% while a basket of diversified stocks returned an average of 19.63%. The standard deviations were 20.39% and 12.85% respectively. On average, did these two investment vehicles produce significantly different returns? Test at the 0.05 level of significance

Answer 1

The provided sample means are shown below:

= 13.37 = 19.63

Also, the provided population standard deviations are:

= 20.39 = 12.85

and the sample sizes are n1 = 100 and n2 = 100.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ1​ = μ2​

Ha: μ1 ​̸​= μ2​

This corresponds to a two-tailed test, for which a z-test for two population means, with known population standard deviations will be used.

(2) Rejection Region

Based on the information provided, the significance level is α = 0.05, and the critical value for a two-tailed test is z_c = 1.96z

(3) Test Statistics

The z-statistic is computed as follows:

z = -2.597

(4) Decision about the null hypothesis

Since it is observed that |z| = 2.597 > z_c = 1.96 , it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0.0094 , and since p = 0.0094 < 0.05 , it is concluded that the null hypothesis is rejected