Question

Obtain the power of your test in part (b) if actually β₁ = 2.0. Assume σ{b₁} = 0.50.

a) 0.50
b) 0.25
c) 0.75
d) 1.0

Answer 1

The power of the test when β₁ is 2.0 with a standard error of 0.50 is approximately 1.0, implying a very high probability of correctly rejecting the null hypothesis.

Step-by-step explanation:

To calculate the power of the test when β₁ is actually 2.0 and the standard error of the estimate, σ{b₁}, is 0.50, we use the formula power = 1 - ß.

First, we need to determine the z-score for the actual value of β₁:

Assuming a two-tailed test and a typical significance level (α), we would reference a standard normal distribution table to find the probability that a z-score of 4 or more extreme occurs by chance. However, since a z-score of 4 is significantly far from the mean, we can safely assume that this probability is almost 0, corresponding to a Type II error rate (ß) of almost 0.

The power of the test, therefore, can be calculated as:

Thus, the power is extremely high, which indicates the test has a very good chance of correctly rejecting the null hypothesis when β₁ is indeed 2.0.