Quality of Tests (Edexcel A Level Further Maths: Further Statistics 1): Exam Questions

A supplier delivers eggs to a shop and claims that no more than 4% of the eggs are cracked. The shop manager suspects that the proportion of cracked eggs is greater than this and carries out a test.

The proportion of cracked eggs is denoted by $p$.

State suitable null and alternative hypotheses for the manager's test.

In test A, the manager takes a single random sample of 80 eggs. Let $X$ be the number of cracked eggs in the sample. The manager rejects the null hypothesis if $X\ge 7$.

Find the size of test A.

In test B, the manager takes a random sample of 50 eggs. Let $Y$ be the number of cracked eggs in this sample. The manager applies the following rules:

• if $Y\ge 6$, reject the null hypothesis;

• if $Y\le 4$, do not reject the null hypothesis;

• if $Y=5$, take a second random sample of 50 eggs and reject the null hypothesis if at least one egg in the second sample is cracked, otherwise do not reject the null hypothesis.

Find the size of test B.

Given that the actual proportion of cracked eggs is 0.10,

(i) find the power of test A,

(ii) find the expected number of eggs sampled using test B.

The manager wants to use the more efficient test. Advise the manager which test to use, giving a reason.