Question

Weighted alpha is a measure of risk-adjusted performance of stocks over a period of a year. A high positive weighted alpha signifies a stock whose price has risen while a small positive weighted alpha indicates an unchanged stock price during the time period. Weighted alpha is used to identify companies with strong upward or downward trends. The weighted alpha for the top $30$ stocks of banks in the northeast and in the west as identified by Nasdaq on May $24,2013$ are listed in Table $10.6$ and Table$10.7,$ respectively

Is there a difference in the weighted alpha of the top $30$ stocks of banks in the northeast and in the west? Test at a $5\%$ significance level. Answer the following questions:

a. Is this a test of two means or two proportions?

b. Are the population standard deviations known or unknown?

c. Which distribution do you use to perform the test?

d. What is the random variable?

e. What are the null and alternative hypotheses? Write the null and alternative hypotheses in words and in symbols.

f. Is this test right, left, or two tailed?

g. What is the p-value?

h. Do you reject or not reject the null hypothesis?

i. At the ___ level of significance, from the sample data, there ______ (is/is not) sufficient evidence to conclude that ______.

j. Calculate Cohen’s d and interpret it

Short answer

(a) Two means.

(b) Unknown.

(c) Student's-t.

(d) The random variable is $\overline{X_1}-\overline{X_2}$.

(e) $$\begin{gathered} H_0:{\mu }_1={\mu }_2 \\ {H}_a:{\mu }_1\ne {\mu }_2 \end{gathered}$$

(f) Two-tailed.

(g) The p-value is $0.8787$.

(h) Do not reject the null hypothesis.

(i) At the $5\%$level of significance, from the sample data, there is not sufficient evidence to conclude that the mean weighted alphas for the banks in the northeast and the west are different.

(j)$d=0.040$, very small, because$0.040$is less than Cohen's value of$0.2$for small effect size.

Step-by-step solution

Given information

Given the weighted alpha for the top $30$stocks of banks in the northeast and in the west on May $24,2013.$

Explanation (part a)

This is a test of two means.

Explanation (part b)

The population standard deviations are unknown.

Explanation (part c)

The distribution used to perform this test is Student's-t.

Explanation (part d)

The random variable $\overline{X_1}-\overline{X_2}$is the difference in weighted alpha between the top $30$ stocks banks in the northeast and the top $30$stocks banks in the west.

As a result, $\overline{X_1}-\overline{X_2}$is the random variable.

Explanation (part e)

The null and hypothesis: $H_0$: The means of the weighted alphas are equal.

The alternative hypothesis: $H_a:$The means of the weighted alphas are not equal.

That is

$$\begin{gathered} H_0:{\mu }_1={\mu }_2 \\ {H}_a:{\mu }_1\ne {\mu }_2 \end{gathered}$$

Explanation (part f) 

The test is two tailed.

Explanation (part g)

The p-value is$0.8787$.

Explanation (part h)

Since $p-value>$significant value, do not reject the null hypothesis.

Explanation (part i)

At the $5\%$ level of significance, from the sample data, there is not sufficient evidence to conclude that the mean weighted alphas for the banks in the northeast and the west are different.

Explanation (part j)

Because $0.040$ is less than Cohen's estimate of $0.2$ for a modest effect size, the effect size is very small$(d=0.040).$ The difference in the means of the weighted alphas for the two bank regions is minor, indicating that there is no substantial difference in their stock patterns.