S4 June 2016 Q3
3. A jar contains a large number of sweets which have either soft centres or hard centres. The jar is thought to contain equal proportions of sweets with soft centres and sweets with hard centres. A random sample of 20 sweets is taken from the jar and the number of sweets with hard centres is recorded.
(a) Using a 5% level of significance, find the critical region for a two-tailed test of the hypothesis that there are equal proportions of sweets with soft centres and sweets with hard centres in the jar. (2)
(b) Calculate the probability of a Type I error for this test. (2)
Given that there are 3 times as many sweets with soft centres as there are sweets with hard centres,
(c) calculate the probability of a Type II error for this test. (2)
| Scheme | Marks |
|---|---|
| \(X\) = No of soft centres. \(X \sim \mathrm{B}(20, 0.5)\) | |
| Critical region \(X \leqslant 5\) or \(X \geqslant 15\) | B1B1 |
| (2) |
Notes
B1 \(X \leqslant 5\)
B1 \(X \geqslant 15\)
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(\text{Type I error}) = \mathrm{P}(X \leqslant 5 \mid p = 0.5) + \mathrm{P}(X \geqslant 15 \mid p = 0.5)\) | M1 |
| \(= 0.0207 + 0.0207 = 0.0414\) | A1 |
| (2) |
Notes
M1 Adding their two CR together or a correct answer
A1 awrt 0.0414
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(\text{Type II error}) = \mathrm{P}(X \lt 15 \mid p = 0.25) - \mathrm{P}(X \lt 6 \mid p = 0.25)\) | M1 |
| \(= 1 - 0.6172 = 0.3828\) | A1 |
| (2) | |
| (6 marks) |
Notes
M1 FT their CR
A1 awrt 0.383