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.

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.