Question

An artist experimenting with clay to create pottery with a special texture has been experiencing difficulty with these special pieces. About 40% break in the kiln during firing. Hoping to solve this problem, she buys some more expensive clay from another supplier. She plans to make and fire 10 pieces and will decide to use the new clay if at most one of them breaks.a) Suppose the new, expensive clay really is no better than her usual clay. What’s the probability that this test convinces her to use it anyway? (Hint: Use a Binomial model.)b) If she decides to switch to the new clay and it is no better, what kind of error did she commit?c) If the new clay really can reduce breakage to only 20%, what’s the probability that her test will not detect the improvement?d) How can she improve the power of her test? Offer at least two suggestions.

Answer 1

Using a binomial model, the probability of the test convincing the artist to use the new clay, even if it is no better, is approximately 38.2%. If she decides to switch to the new clay and it is no better, she would commit a type II error. If the new clay can reduce breakage to 20%, the probability that her test will not detect the improvement is approximately 67.4%. To improve the power of her test, she could increase the sample size and/or decrease the significance level.

Step-by-step explanation:

To solve this problem, we can use a binomial model. Let's define success as a piece not breaking during firing. The probability of success for the usual clay is 60% (since 40% break). Assuming the new clay is equally as likely to break, the probability of success for the new clay is also 60%. The artist plans to make and fire 10 pieces, so for her to decide to use the new clay, at most one of them should break. The probability of at most one success can be calculated using the binomial probability formula:

P(X ≤ 1) = C(10, 0) * (0.6)^0 * (0.4)^10 + C(10, 1) * (0.6)^1 * (0.4)^9

After solving this equation, the probability of the test convincing her to use the new clay, even if it is no better, is approximately 0.382 or 38.2%.

b) If she decides to switch to the new clay and it is no better, she would commit a type II error. This means she would fail to reject the null hypothesis (that the new clay is no better) when it is actually true.

c) If the new clay can reduce breakage to 20%, the probability that her test will not detect the improvement (type II error) can be calculated using the binomial probability formula with a success probability of 20%. The probability of at most one success would be:

P(X ≤ 1) = C(10, 0) * (0.2)^0 * (0.8)^10 + C(10, 1) * (0.2)^1 * (0.8)^9

After solving this equation, the probability of not detecting the improvement is approximately 0.674 or 67.4%.

d) To improve the power of her test, the artist could:

- Increase the sample size: By making and firing more than 10 pieces, she can reduce the uncertainty and increase the power of her test.

- Decrease the significance level: If she is willing to accept a higher chance of making a type I error (wrongly concluding that the new clay is better), she can decrease the significance level (typically set at 0.05) to increase the power of her test.

Answer 2

Step-by-step explanation:

a. If the new clay has the same probability of failing as her usual clay, then the probability of a piece breaking is p = 0.40, and the probability of it not breaking is q = 0.60.

There are 10 pieces. The probability that at most 1 fails (0 fail or 1 fails) is found with binomial probability:

P = ₁₀C₀ (0.40⁰) (0.60¹⁰) + ₁₀C₁ (0.40¹) (0.60⁹)

P = 0.046

There is a 4.6% chance that at most 1 piece breaks, convincing her to use the new clay.

b. This is a Type I error (also known as a false positive).

c. If p is reduced to 0.20, and q is 0.80, then the probability that more than 1 breaks is:

P = 1 − ₁₀C₀ (0.20⁰) (0.80¹⁰) − ₁₀C₁ (0.20¹) (0.80⁹)

P = 0.624

There is a 62.4% probability that more than 1 piece will break, causing her to reject the new clay.

(This would be a Type II error, or a false negative).

d. She can increase the power of her test in two ways. The first is by increasing the number of pieces she tests (for example 20 pieces instead of 10). This will decrease the probability of making a Type I error.

The second way is by increasing the number of pieces that can break before she rejects the clay (for example, at most 2 instead of at most 1). This will decrease the probability of making a Type II error.