Final answer:
To find the power of the test, use the critical value to find the approximate value for beta, then subtract beta from 1 to find the approximate power.
Step-by-step explanation:
To find the power of the test, we first need to determine the critical value, which is the value of standard deviation that we can reject the null hypothesis. We use the level of significance, alpha=0.05, to find this value. The critical value is the value that separates the rejection region from the non-rejection region. In this case, since we are testing if the true population standard deviation is 6.0, we want to find the critical value that corresponds to a sample standard deviation of 6.0. We can use a standard normal distribution table or a calculator to find the critical value. For alpha=0.05, the critical value is approximately 1.96.
Once we have the critical value, we can find the approximate value for beta, which is the probability of making a Type II error, or failing to reject the null hypothesis when it is false. The power of the test is equal to 1 - beta. To find beta, we need the standard deviation under the alternative hypothesis, which is 6.0. Using the critical value of 1.96, we can calculate the z-score corresponding to a standard deviation of 6.0. The z-score is equal to (sample standard deviation - hypothesized standard deviation) / (standard error) = (6.0 - 3.0) / (3.0 / sqrt(20)) = 3.0 / (3.0 / sqrt(20)) = (3.0 * sqrt(20)) / 3.0 = sqrt(20) = approximately 4.47. From the standard normal distribution table or a calculator, we can find the probability to the left of 4.47, which is the probability of making a Type II error. The beta is equal to this probability.
Finally, to find the approximate power, we subtract beta from 1. The power is equal to 1 - beta. In this case, the power is equal to 1 - beta = 1 - probability to the left of 4.47 = 1 - approximately 0.99999 = approximately 0.00001.