mu=78

population std deviation=12
Normal Distribution
n=16
alpha level=.05

* compute the power of the hypothesis test if the program has a 3-point effect.
I came up with 83.88 %

* compute the power of the hypothesis test if the program has a 6-point effect.

I looked at the example in the book and I'm just not understanding! Help. Thanks.

To compute the power of a hypothesis test, you need to follow a certain procedure. In this case, we have the following information:

Mean under the null hypothesis (mu): 78
Population standard deviation (sigma): 12
Sample size (n): 16
Alpha level (α): 0.05

To compute the power of the hypothesis test, we need to determine the effect size, which is the difference between the mean under the alternative hypothesis and the mean under the null hypothesis. In this case, you mentioned that there is a 3-point effect in the first scenario and a 6-point effect in the second scenario.

Now, let's calculate the power for each scenario:

Scenario 1: 3-point effect
Under the alternative hypothesis, the mean would be μ + 3 = 78 + 3 = 81.

To compute the power, we need to calculate the Z-score corresponding to the value of the alternative mean.

Z1 = (mu - μ) / (sigma / sqrt(n))
= (81 - 78) / (12 / sqrt(16))
= 3 / (12 / 4)
= 3 / 3
= 1

Next, we need to find the area under the standard normal distribution curve to the right of Z1. In other words, we need to find the probability that Z is greater than or equal to 1. We can use a Z-table or a calculator to find the corresponding probability.

Using a Z-table, the probability is approximately 0.1587.

Therefore, the power of the hypothesis test with a 3-point effect is 1 - 0.1587 = 0.8413 or 84.13%.

Scenario 2: 6-point effect
Using the same procedure as before, we find:

Z2 = (mu - μ) / (sigma / sqrt(n))
= (84 - 78) / (12 / sqrt(16))
= 6 / (12 / 4)
= 6 / 3
= 2

Again, we need to find the area to the right of Z2, which is the probability that Z is greater than or equal to 2.

Using a Z-table, the probability is approximately 0.0228.

Therefore, the power of the hypothesis test with a 6-point effect is 1 - 0.0228 = 0.9772 or 97.72%.

In summary, the power of the hypothesis test is higher when the effect size is larger. In the first scenario, with a 3-point effect, the power is 84.13%. In the second scenario, with a 6-point effect, the power is 97.72%.