Question

The "beta coefficient" of a stock is a measure of the stock`s volatility (or risk) relative to the market as a whole. Stocks with beta coefficients greater than 1 generally bear greater risk (more volatility) than the market, whereas stocks with beta coefficients less than 1 are less risky (less volatile) than the overall market. A random sample of 15 high-technology stocks was selected at the end of 2009, and the sample mean and sample standard deviation of the beta coefficients were computed to be 1.23 and s=.37, respectively. (Assume that beta coefficient follows a normal distribution.)

a. (4 points) State the null and alternative hypothesis to test whether the average high-technology stock is riskier than the market as a whole.
b. (2 points) Establish the appropriate test statistics for this problem.
c. (4 points) Determine an appropriate decision rule for testing the hypothesis in part a at the level of significance α = .01.
d. (6 points) Make an appropriate conclusion based on the summary statistics provided in the problem and the decision rule you specified in part c.
e. (3 points) What is the p-value of the test.
f. (2 points) Based on the p-value calculated in part (e), do you reject the null hypothesis at the level of significance α=.01? Say how you made your decision.
g. (3 points) How does your conclusion change if instead of α=.01 we chose a smaller level of significance? Explain.

Answer 1

a)
Below are the null and alternative Hypothesis,
Null Hypothesis, H0: μ = 1
Alternative Hypothesis, Ha: μ > 1

b)
Test statistic,
t = (xbar - mu)/(s/sqrt(n))
t = (1.23 - 1)/(0.37/sqrt(15))
t = 2.408

c)
Rejection Region
This is right tailed test, for α = 0.01 and df = 14
Critical value of t is 2.624.
Hence reject H0 if t > 2.624

d)
Failed to reject null hypothesis

e)
P-value = 0.0152

f)
Failed to reject the null hypothesis as p-value is greater than 0.01

g)
There is no change in the conclusion on decreasing the alpha because p-value will be greater than alpha and we will conclude to reject the null hypothesis.