A manufacturer of rotors for drones claims that at most 3% of its rotors are def

Question

A manufacturer of rotors for drones claims that at most 3% of its rotors are defective.

(a) A random sample of 24 rotors is selected and it is found that 2 of them are defective. Is it fair to reject the manufacturer's claim based on this observation?

(b) A random sample of 24 rotors is selected and it is found that 4 of them are defective. Now is it fair to reject the manufacturer's claim based on this observation?

Hint: Be careful, this is NOT a question about the probably of exactly 2 or exactly 4 being defective. What is really being asked is the following. Suppose X = "the number of defective rotors in a shipment of N rotors, with the assumption that the probability of any individual rotor being defective is 0.03." Is this binomial distribution a reasonable model for the company's quality control efforts? When K rotors are found to be defective, what we are really asking is: "is this level of defective rotors, or worse, consistent with the model proposed by the company, where they said it is at most 3%"? So we are interested in the probability of P(X K). If this probability is very small, then something very, very unusual just happened when we tested the company's shipment, and we probably want to reject the claim. If the probability is not small, then the model may be reasonable. This paradigm is called "hypothesis testing" and it is essentially a probabalistic version of "proof by contradiction" (assume the claim is true, and then derive a contradiction, proving that the claim is false; here we prove that if the assumption leads to a conclusion with a very small probability, then the original assumption was probably not true).

Explanation / Answer

a) Sample of 24 rotors is selcted and 2 of them are defective = 2

Null Hypothesis : H0 : At most 3% of rotors are defective. COmpany's claim is correct. p <= 0.03

Alternative Hypothesis : Ha : More than 3% of rotors are defective. Company's claim is correct. p > 0.03

As sample size is small, we can use binomial test here instead of normal approximation to binomial.

Here p - value = Pr(X >= 2; 24; 0.03) = 1 - Pr(X <2; 24; 0.03)

= 1 - 24C1 (0.03)(0.97)23 -  24C0 (0.97)24

= 1 - 0.3573 - 0.4814

= 0.1613

so, there are 16.13% probability that 2 or more rotors can be faulty in the batch. So, we can accept the claim of company at 0.05 statistically significance level.

(b) Now , 4 rotors are faulty

Here p - value = Pr(X >= 4; 24; 0.03) = 1 - Pr(X <3; 24; 0.03)

= 1 - 24C1 (0.03)(0.97)23 -  24C0 (0.97)24  - 24C2(0.97)22 (0.03)2 - 24C3 (0.97)21 (0.03)3

= 1 - 0.9947

= 0.0053

so there is only 0.53% probability of that thing occuring. That now we shall reject the null hypothesis and can say that company claim is false.

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