Question

Two different types of injection-molding machines are used to form plastic parts. A part is considered defective if it has excessive shrinkage or is discolored. Two random samples, each of size 300, are selected, and 15 defective parts are found in the sample from machine 1, while 8 defective parts are found in the sample from machine 2. Suppose that p1 = 0.05 and p2 = 0.01.(a) With the sample sizes given, what is the power of the test for this two sided alternative? Power =(b) Determine the sample size needed to detect this difference with a probability of at least 0.9. Use α = 0.05. n =

Answer 1

The power of the test is the probability that the test will correctly reject a null hypothesis. To calculate the power of the test for a two-sided alternative, we need to determine the critical value and the standard deviation. The sample size needed to detect a difference with a probability of at least 0.9 can be calculated using the formula for sample size determination.

Step-by-step explanation:

The power of a statistical test is the probability that the test will correctly reject a null hypothesis when the alternative hypothesis is true. In this case, the alternative hypothesis is that there is a difference in the proportion of defective parts between the two machines. To calculate the power of the test, we need to determine the critical value and the standard deviation.

Step 1: Calculate the critical value

We need to determine the critical value based on the significance level and the alternative hypothesis. Since it is a two-sided test, the critical value is calculated using the z-score.

Critical value = Z(α/2) where α = 0.05

Z(0.025) ≈ 1.96

Step 2: Calculate the standard deviation

The standard deviation is calculated using the formula:

Standard deviation = √(p1(1-p1)/n1 + p2(1-p2)/n2)

Standard deviation = √((0.05(1-0.05)/300) + (0.01(1-0.01)/300))

Step 3: Calculate the power of the test

The power of the test is calculated using the formula:

Power = Φ[(p1-p2)/standard deviation - critical value]

Power = Φ[(0.05-0.01)/standard deviation - 1.96]

Step 4: Calculate the sample size needed

To determine the sample size needed to detect the difference with a probability of at least 0.9, we can use the formula:

n = ((Z(1-α/2)+Z(1-ß))^2) (p(1-p))/(p1-p2)^2

We assume that p = (p1+p2)/2

Answer 2

To calculate the power of the test, use the formula Power = 1 - (1 - p1)^n1 * (1 - p2)^n2. To determine the sample size needed for a desired probability, use the formula n = (Zα + Zß)^2 * (p1 * (1 - p1) + p2 * (1 - p2)) / (p1 - p2)^2.

Step-by-step explanation:

To calculate the power of the test for a two-sided alternative, we can use the formula:

Power = 1 - (1 - p1)^n1 * (1 - p2)^n2

Substituting the given values:

Power = 1 - (1 - 0.05)^300 * (1 - 0.01)^300

Solving this equation will give us the power of the test.

To determine the sample size needed to detect this difference with a probability of at least 0.9, we can use the formula:

n = (Zα + Zß)^2 * (p1 * (1 - p1) + p2 * (1 - p2)) / (p1 - p2)^2

Substituting the given values:

n = (1.96 + 0.842)^2 * (0.05 * (1 - 0.05) + 0.01 * (1 - 0.01)) / (0.05 - 0.01)^2

Solving this equation will give us the required sample size.