1. ## Binomial Distribution Question

Dear all,

I have stumbled upon this impossible question from my teacher. The question is a binomial distribution and is as follows:

. A bakery supplies batches of jam doughnuts for a local supermarket. Batches are either high-quality or low-quality. 1% of doughnuts in high-quality batches have no jam. 10% of doughnuts in low-quality batches have no jam. To try to intercept low-quality batches, the supermarket takes a sample of 50 doughnuts from each batch and tests them. If two or more doughnuts are found to have no jam, the batch is rejected.

(a) Explain what a Type I error and Type II error would be in this case.

(b) Calculate the probability of each type of error.

Hint: use the binomial distribution: pr(r) = NCr pr(1-p)N-r where r is the number of times a particular outcome occurs in N trials and p is the probability of that outcome occurring in 1 trial

(c) Calculate the power of the test.

(d) The supermarket can alter the error probabilities by changing the sample size or by changing the rejection criteria. What would you recommend if: (i) doughnuts without jam are found to affect consumer safety? (ii) there is no alternative bakery that could supply doughnuts?

Now I can easily solve for (a), but (b) is just killing me, I always end up with 0.006% from (0.99^48 * 0.01 *0.01), the teacher hinted 8% at me but she did not confirm it and I may just have heard her wrong. I cant really solve this question and its killing me!!!

2. Originally Posted by Hashem
Dear all,

I have stumbled upon this impossible question from my teacher. The question is a binomial distribution and is as follows:

. A bakery supplies batches of jam doughnuts for a local supermarket. Batches are either high-quality or low-quality. 1% of doughnuts in high-quality batches have no jam. 10% of doughnuts in low-quality batches have no jam. To try to intercept low-quality batches, the supermarket takes a sample of 50 doughnuts from each batch and tests them. If two or more doughnuts are found to have no jam, the batch is rejected.

(a) Explain what a Type I error and Type II error would be in this case.

(b) Calculate the probability of each type of error.

Hint: use the binomial distribution: pr(r) = NCr pr(1-p)N-r where r is the number of times a particular outcome occurs in N trials and p is the probability of that outcome occurring in 1 trial

(c) Calculate the power of the test.

(d) The supermarket can alter the error probabilities by changing the sample size or by changing the rejection criteria. What would you recommend if: (i) doughnuts without jam are found to affect consumer safety? (ii) there is no alternative bakery that could supply doughnuts?

Now I can easily solve for (a), but (b) is just killing me, I always end up with 0.006% from (0.99^48 * 0.01 *0.01), the teacher hinted 8% at me but she did not confirm it and I may just have heard her wrong. I cant really solve this question and its killing me!!!

$\alpha = \Pr(X \geq 2 \, | \, \text{High quality}) = \Pr(X \geq 2 \, | \, X$ ~ Binomial(n = 50, p = 0.01) $)$.
$\beta = \Pr(X < 2 \, | \, \text{Low quality}) = \Pr(X < 2 \, | \, X$ ~ Binomial(n = 50, p = 0.1) $)$.