In each batch of manufactured articles, $5\%$ of the articles are found to be defective. A quality

Question

In each batch of manufactured articles, $5\%$ of the articles are found to be defective. A quality inspection is carried out by checking samples of $20$ articles. If $2$ or fewer defective articles are found in the sample of $20$, the batch is accepted. Find the probability that the batch is accepted.[1] Find the least value of $n$ such that the probability of having less than $n$ defective articles in a sample of $20$ articles is greater than $0.99$.[2] Fifty random samples of $20$ articles each were taken. Find the probability that the average number of defective articles per sample is at most $1.25$.[3] It is proposed that a smaller sample be taken for inspection. Find the largest value of $k$ such that the probability of having at least $1$ defective article in a sample of $k$ articles is to be less than $0.4$?[3]

Accepted Answer

(i) Required probability ${\approx 0.925}$ (ii) Least value of ${n=5}$ (iii) ${\mathrm{P}\left(\overline{Y} \leq 1.25\right) \approx 0.965}$ (iv) largest value of ${k=9}$

2022 VJC P2 Q9

2022 VJC P2 Q9

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