Chapter 14: Statistical hypothesis testing
The chapter problem: is the company's claim believable?
A snack company prints one line on every packet: "1 in 4 packets contains a voucher." The claim is about every packet the company makes. A customer buys 20 packets and finds 1 voucher.
The customer cannot open every packet in the factory. The 20 packets are a sample, and the single voucher is the evidence. The question is whether 1 voucher in 20 is strong enough to doubt the claim, or whether a run like this happens by chance.
By the end of this chapter you will answer that question with a number, not a feeling. You will measure how surprising the evidence is, and you will say how sure you are.
First contact — no working needed yet. If 1 in 4 packets really does contain a voucher, how many vouchers would you expect in 20 packets? Write that number down. Then decide whether finding 1 voucher would surprise you. Write one sentence saying why. Keep both answers for the Review.
A second prediction: write down what you would conclude if the customer had found 5 vouchers instead of 1.
If you are not sure where to start
A proportion of 1 in 4 means one quarter. One quarter of 20 is 5, so the expected count is 5. The count you actually see will vary from sample to sample, so a count near 5 is ordinary and a count far from 5 is surprising. For the second prediction, 5 vouchers is exactly the expected count. It cannot be evidence against the claim.
By the end of this chapter you will be able to
- write a null hypothesis and an alternative hypothesis for a claim about a proportion;
- choose a one-tailed or a two-tailed test, and explain the choice;
- explain the significance level as the probability of rejecting the null hypothesis when it is true;
- name and use the test statistic, the critical value, the critical region, the acceptance region and the p-value;
- carry out a binomial hypothesis test for a proportion, and interpret the result in the context of the original claim;
- carry out a hypothesis test for the mean of a Normal distribution when the variance is known, given or assumed;
- interpret a given correlation coefficient using a given p-value or a critical value, and explain why a correlation is not a cause.
Where it is used
Every serious claim is checked against evidence. A drug trial asks whether a new treatment works better than the old one. A factory asks whether a machine has drifted away from its target weight. A polling company asks whether one candidate is really ahead. A scientist asks whether a result is real or the product of chance.
In each case the claim is a statement about a whole population, and the evidence is a sample. A hypothesis test turns the gap between the claim and the sample into a probability. That probability tells you how much weight the evidence can carry.
The test never proves a claim. It measures how well the claim survives the evidence, and it says how often a test like this gives the wrong answer.
Your turn to connect it
Think of a number you have seen in an advert, a news report or a school notice: "9 out of 10 owners", "our average waiting time is 4 minutes", "twice as many people prefer this brand". Write the claim down.
Then write two things. First, what sample would you collect to check it? Second, what result would make you doubt the claim? Keep your answer for the Review.