Bulb reliability
A supplier claims 5 % of its bulbs are faulty, and a test finds 4 in 20: binomial tails as the first step towards a hypothesis test.
Draws on 5. The binomial distribution, 11. Student's t and confidence intervals
Step 1The claim
A supplier states that 5 % of its LED bulbs are faulty. A buyer tests 20 bulbs from a delivery and finds 4 faulty. Is that bad luck, or evidence that the delivery is worse than claimed?
Step 2What the claim predicts
If the claim is true, the number of faulty bulbs in 20 is \(X \sim B(20, 0.05)\), with a mean of 1. No faulty bulb at all is the likeliest single outcome, with probability 0.3585. The question is how often the claim would produce a result at least as bad as the one found.
Four or more faulty
If 5 % of bulbs are faulty, what is the probability of finding 4 or more faulty bulbs among 20?
Step 3How many would be surprising?
A fair judgement fixes in advance how rare a result must be before it casts doubt on the claim, say a chance of 5 % or less, and only then looks at the count.
Where doubt begins
If 5 % of bulbs are faulty, what is the smallest number k of faulty bulbs among 20 for which finding k or more has a probability of at most 0.05?
Step 4The same share in a bigger sample
Two faulty bulbs in 20 is 10 %, twice the claim, yet \(P(X \ge 2) = 0.2642\): nothing unusual. The same 10 % in a test of 100 bulbs is another matter.
Ten in a hundred
If 5 % of bulbs are faulty, what is the probability of finding 10 or more faulty bulbs among 100?
Step 5From here to a hypothesis test
This case has run a hypothesis test in all but name. Assume the claim (the null hypothesis); compute how likely a result at least as extreme as the one observed would be (the p-value, 0.0159 for 4 of 20); and doubt the claim when that is below a level fixed in advance, such as 0.05. The Hypothesis Testing Lab, the next lab in this series, makes each step precise, with the errors a test can make and the power to detect a real problem.