S4 January 2006 Q2
2.
(a) Define
(i) a Type I error, (1)
(ii) a Type II error. (1)
A manufacturer sells socks in boxes of 50.
The mean number of faulty socks per box is 7.5. In order to reduce the number of faulty socks a new machine is tried. A box of socks made on the new machine was tested and the number of faulty socks was 2.
(b)
(i) Assuming that the number of faulty socks per box follows a binomial distribution derive a critical region needed to test whether or not there is evidence that the new machine has reduced the mean number of faulty socks per box. Use a 5% significance level. (2)
(ii) Stating your hypotheses clearly, carry out the test in part (i). (3)
(c) Find the probability of the Type I error for this test. (2)
(d) Given that the true mean number of faulty socks per box on the new machine is 5, calculate the probability of a Type II error for this test. (3)
(e) Explain what would have been the effect of changing the significance level for the test in part (b) to 2½%. (1)
| Scheme | Marks |
|---|---|
| (i) Type I: \(\mathrm{H}_0\) rejected when true | B1 |
| (ii) Type II: \(\mathrm{H}_0\) accepted when false | B1 |
| (2) |
| Scheme | Marks |
|---|---|
| (i) \(p = \dfrac{7.5}{50} = 0.15\) | B1 |
| cr \(X \leqslant 3\) | B1 |
| (2) | |
| (ii) \(\mathrm{H}_0 : p = 0.15,\quad \mathrm{H}_1 : p \lt 0.15\) both | B1 |
| \(x = 2\) in cr \(X \leqslant 3\) so \(\mathrm{H}_0\) is rejected | M1 |
| The new machine has reduced the mean number of faulty socks | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(\text{Type I error}) = \mathrm{P}(X \leqslant 3 \mid p = 0.15) = 0.0460\) | M1 A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(\text{Faulty}) = \dfrac{5}{50} = 0.1\) | B1 |
| \(\mathrm{P}(\text{Type II error}) = \mathrm{P}(X \geqslant 4 \mid p = 0.1) = 1 - 0.2503\) | M1 |
| \(= 0.7497\) awrt 0.750 | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| Critical region changes to \(X \leqslant 2\). \(\mathrm{H}_0\) still rejected. | B1 |
| (1) | |
| (13 marks) |