S2 January 2009 Q3
3. A single observation \(x\) is to be taken from a Binomial distribution \(\mathrm{B}(20, p)\).
This observation is used to test \(\mathrm{H}_0 : p = 0.3\) against \(\mathrm{H}_1 : p \ne 0.3\)
(a) Using a 5% level of significance, find the critical region for this test. The probability of rejecting either tail should be as close as possible to 2.5%. (3)
(b) State the actual significance level of this test. (2)
The actual value of \(x\) obtained is 3.
(c) State a conclusion that can be drawn based on this value giving a reason for your answer. (2)
| Scheme | Marks |
|---|---|
| \(X \sim \mathrm{B}(20, 0.3)\) | M1 |
| \(\mathrm{P}(X \leqslant 2) = 0.0355\) \(\mathrm{P}(X \geqslant 11) = 1 - 0.9829 = 0.0171\) | |
| Critical region is \((X \leqslant 2) \cup (X \geqslant 11)\) | A1 A1 |
| (3) |
| Scheme | Marks |
|---|---|
| Significance level \(= 0.0355 + 0.0171,\ = 0.0526\) or 5.26% | M1 A1 |
| (2) |
| Scheme | Marks |
|---|---|
| Insufficient evidence to reject \(\mathrm{H}_0\) Or sufficient evidence to accept \(\mathrm{H}_0\)/not significant | B1 ft |
| \(x = 3\) (or the value) is not in the critical region or \(0.1071 \gt 0.025\) | B1 ft |
| (2) | |
| (7 marks) |
Notes
Do not allow inconsistent comments