You are a successful entrepreneur that has developed a new sustainable product that is manufactured through a standard production process. As part of this process, the product goes through quality assurance testing. The quality assurance inspector selects 10 units of your product at random from the production batch; if at least 3 units are defective, then the batch is rejected and will not be sold in stores. Previous records show that 10% of the products in each batch have issues.
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What is the mean number of defective units in the sample?
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What is the standard deviation of defective units in the sample?
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What is the probability that the batch will be rejected?
I'm having a little trouble calculating the mean and standard deviation.
However, what I have got so far was that the mean is 1, standard deviation is 3, then the probability would be 0.2514.
I'm very unsure on my answer and would like to see if anyone can point out anything I'm doing wrong.
Thanks a lot! 🙂
Best Answer
I'm assuming a very large population of units from which to select, so that the distribution is binomial, and that 'have issues' means 'are defective." Then we have the number of defective units seen $X \sim Bin(n=10, p=.1)$ Thus, (1) $E(X) = np = 1$ as you say. (2) V(X) = np(1-p) = .9, so SD(X) = \sqrt(.9) = 0.9487.$; I think you overlooked a decimal point.
(3) $P\{Rej\} = P\{X \ge 3\} = 1 - P\{X \le 2\} = 0.0701,$ where $P\{X \le 2\}$ is the sum of three terms, the first of which is 0.3487. Use the formula for individual binomial probabilities (usually called PMF or PDF) to fill in the gaps in the computation. Alternatively, see if your book has a binomial table for $n = 10, p=.01.$ or use software. In R statistical software, you can get the answer from code '1 - pbinom(2, 10, .1)'.