Chapter 14: Review

The chapter problem, solved

The snack company claims that 1 packet in 4 contains a voucher. A customer buys 20 packets and finds 1 voucher. Decide whether the claim is believable at the 5% level.

Step 1. Write the hypotheses and the level

The claim is the null hypothesis. The customer suspects too few vouchers, so the test is one-tailed on the lower tail:

Step 2. State the model under $H_0$

Under the claim, each packet contains a voucher with probability $0.25$. The number of vouchers in 20 packets is

The model assumes the packets are independent and that each has the same probability. Both assumptions are checked at the end.

Step 3. Find the test statistic

The sample gives

Step 4. Find the p-value and the critical region

The lower tail contains the observed value and every smaller count:

Work up the tail to find the critical region. The next value gives

so the critical region at the 5% level is $X \le 1$, with critical value 1. The acceptance region is $X \ge 2$.

Step 5. Compare, decide and interpret

so the test rejects $H_0$. The value $X = 1$ lies in the critical region, which agrees with the p-value route.

There is sufficient evidence at the 5% level that fewer than 1 packet in 4 contains a voucher.

What the test does and does not show

The sample proportion is $\dfrac{1}{20} = 0.05$, far below the claimed $0.25$. If the claim were true, a result this low would happen in only about 2.4% of samples of 20. That is the evidence against the claim.

The test does not prove that the claim is false. It says that the evidence is strong enough to doubt the claim at the 5% level of risk.

The level matters. At the 1% level the same p-value gives

so the test would retain $H_0$. A conclusion must always name the level.

Three limits apply to the conclusion.

• The sample is small. Twenty packets give limited information. A larger sample would give a clearer answer.
• The packets may not be independent. If the 20 packets came from one box or one delivery, one missing batch of vouchers could produce the whole result.
• The sample may not be random. The customer chose the shop and the day. Another shop may stock a different mix of packets.

Return to the first-contact predictions. One quarter of 20 is 5, so 5 vouchers were expected. Finding 1 is surprising enough to reject the claim at the 5% level. Finding 5 would have been ordinary, and the test would have retained the claim.

Mixed practice — choose the method first

For each question, say which method you will use before you calculate: a binomial test, a Normal test for a mean, a Normal or binomial probability, a summary statistic, or a comparison with a critical value. The items mix this chapter with Chapters 11, 13 and 4.

1. A company claims that 1 packet in 4 contains a voucher. A customer buys 20 packets and finds 1 voucher. Test the claim at the 5% level against the alternative that fewer packets contain a voucher.
2. A machine fills packets with a mean of 500 g. The standard deviation is known to be 4 g. A sample of 16 packets has mean 497 g. Test at the 5% level whether the mean is below 500 g.
3. A sample of 30 pairs gives $r = 0.42$ and a two-tailed p-value of $0.021$. State, with a symbol, how the p-value compares with the 5% significance level, and name the decision.
4. A binomial variable has $n = 15$ and $p = 0.2$. Find $P(X \le 1)$, to 4 decimal places.
5. A Normal distribution has mean 100 and standard deviation 15. Find $P(X \le 85)$, to 4 decimal places.
6. A sample of six masses, in grams, is $12, 15, 18, 20, 25, 30$. Find the sample mean.
7. A sample is $2, 4, 4, 4, 5, 5, 7, 9$. Find the sample variance, using the sample convention that divides by $n - 1$.
8. A coin is tossed 18 times and shows 14 heads. Test at the 5% level whether the coin is fair.
9. A depot claims that 20% of vans arrive late. In 25 vans, 10 arrive late. Test at the 5% level whether the late rate is higher than the claim.
10. A claim sets $\mu = 20$. The standard deviation is known to be 2. A sample of four items has mean 22. Test at the 5% level whether the mean has changed.
11. A sample of 20 pairs gives $r = -0.61$. The test is two-tailed at the 5% level and the critical value for 20 pairs is $0.444$. State, with a symbol, how $|r|$ compares with the critical value, and name the decision.
12. A sample of 15 pairs gives $r = 0.48$. The test is two-tailed at the 5% level and the critical value for 15 pairs is $0.514$. State, with a symbol, how $|r|$ compares with the critical value, and name the decision.
13. A binomial model has $n = 6$. Work out $\dbinom{6}{3}$.
14. A test gives a p-value of $0.03$. State, with symbols, how this value compares with the 5% level and with the 1% level, and give the decision at each level.

What you can now do

• write a null hypothesis and an alternative hypothesis for a proportion or a mean;
• choose a one-tailed or two-tailed test and give a reason;
• explain the significance level as the probability of rejecting a true null hypothesis;
• carry out a binomial test for a proportion, with the exact p-value or the critical region;
• carry out a Normal test for a mean when the variance is known, given or assumed;
• interpret a correlation coefficient with a given p-value or critical value;
• write a conclusion in context that names the significance level and the risk it carries.

Your relevance reflection

Return to the prompt in the chapter opener: a number you have seen in an advert, a news report or a school notice. Write the claim again.

Now add three things you could not have said before this chapter. First, write the null hypothesis and the alternative for the claim. Second, say what sample would test it and what test you would use. Third, say what a p-value would tell you, and what it would still leave open. Finish with one sentence about how a small sample could make a true claim look false, or a false claim look true.