S4 June 2014 Q2
2.
(a) Define
(i) a Type I error,
(ii) a Type II error. (2)
Rolls of material, manufactured by a machine, contain defects at a mean rate of 6 per roll.
The machine is modified. A single roll is selected at random and a test is carried out to see whether or not the mean number of defects per roll has decreased. The significance level is chosen to be as close as possible to 5%.
(b) Calculate the probability of a Type I error for this test. (3)
(c) Given that the true mean number of defects per roll of material made by the machine is now 4, calculate the probability of a Type II error. (2)
| Scheme | Marks |
|---|---|
| (i) Type I – \(\mathrm{H}_0\) rejected when it is true | B1 |
| (ii) Type II – \(\mathrm{H}_0\) is accepted when it is false | B1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(X \lt c \mid \lambda = 6) \approx 0.05\) | M1 |
| \(\mathrm{P}(X \leqslant 2) = 0.0620\) | |
| \(\mathrm{P}(X \leqslant 1) = 0.0174\) | |
| Critical region \(= X \leqslant 2\) | A1 |
| \(\mathrm{P}(\text{Type 1 error}) = \mathrm{P}(X \leqslant 2 \mid \lambda = 6) = 0.062\) | A1cao |
| (3) |
Notes
M1 use of Po(6)
A1 correct CR. May be implied by correct probability. Alow if written as part of a 2 tailed CR
A1 awrt 0.062
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(\text{Type 2 error}) = \mathrm{P}(X \geqslant 3 \mid \lambda = 4)\) | M1 |
| \(= 1 - 0.2381\) | |
| \(= 0.7619\) | A1 |
| (2) | |
| (7 marks) |
Notes
M1 using Po(4) and \(1 - \mathrm{P}(X \leqslant 2)\), ft their CR in (b) if one tail
A1 awrt 0.762