S4 June 2008 Q6
6. A drug is claimed to produce a cure to a certain disease in 35% of people who have the disease. To test this claim a sample of 20 people having this disease is chosen at random and given the drug. If the number of people cured is between 4 and 10 inclusive the claim will be accepted. Otherwise the claim will not be accepted.
(a) Write down suitable hypotheses to carry out this test. (2)
(b) Find the probability of making a Type I error. (3)
The table below gives the value of the probability of the Type II error, to 4 decimal places, for different values of \(p\) where \(p\) is the probability of the drug curing a person with the disease.
| P(cure) | 0.2 | 0.3 | 0.4 | 0.5 |
| P(Type II error) | 0.5880 | \(r\) | 0.8565 | \(s\) |
(c) Calculate the value of \(r\) and the value of \(s\). (3)
(d) Calculate the power of the test for \(p = 0.2\) and \(p = 0.4\) (2)
(e) Comment, giving your reasons, on the suitability of this test procedure. (2)
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : p = 0.35 \qquad \mathrm{H}_1 : p \neq 0.35\) | B1 B1 |
| (2) |
| Scheme | Marks |
|---|---|
| Let \(X\) = Number cured then \(X \sim \mathrm{B}(20, 0.35)\) | B1 |
| \(\alpha = \mathrm{P}(\text{Type I error}) = \mathrm{P}(x \leqslant 3) + \mathrm{P}(x \geqslant 11)\) given \(p = 0.35\) | M1 |
| \(= 0.0444 + 0.0532\) \(= 0.0976\) | A1 |
| (3) |
| Scheme | Marks | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\beta = \mathrm{P}(\text{Type II error}) = \mathrm{P}(4 \leqslant x \leqslant 10)\) | M1 | ||||||||||
| A1 A1 | ||||||||||
| (3) |
| Scheme | Marks |
|---|---|
| Power \(= 1 - \beta\) | M1 |
| 0.4120 0.1435 | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| Not a good procedure. | B1 |
| Better further away from 0.35 or This is not a very powerful test (power \(= 1 - \beta\)) | B1dep |
| (2) |