S2 January 2010 Q6
6.
A discrete random variable \(X\) has a Binomial distribution \(\mathrm{B}(30, p)\). A single observation is used to test \(\mathrm{H}_0 : p = 0.3\) against \(\mathrm{H}_1 : p \neq 0.3\)
The value of the observation was found to be 15.
| Scheme | Marks |
|---|---|
| The set of values of the test statistic for which | B1 |
| the null hypothesis is rejected in a hypothesis test. | B1 |
| (2) |
Notes
1st B1 for “values/ numbers”
2nd B1 for “reject the null hypothesis” o.e or the test is significant
| Scheme | Marks |
|---|---|
| \(X \sim \mathrm{B}(30, 0.3)\) | M1 |
| \(\mathrm{P}(X \leqslant 3) = 0.0093\) \(\mathrm{P}(X \leqslant 2) = 0.0021\) | A1 |
| \(\mathrm{P}(X \geqslant 16) = 1 - 0.9936 = 0.0064\) \(\mathrm{P}(X \geqslant 17) = 1 - 0.9979 = 0.0021\) | A1 |
| Critical region is \((0 \leqslant)\, x \leqslant 2\) or \(16 \leqslant x\, (\leqslant 30)\) | A1A1 |
| (5) |
Notes
M1 for using B(30,0.3)
1st A1 \(\mathrm{P}(x \leqslant 2) = 0.0021\)
2nd A1 0.0064
3rd A1 for \((X) \leqslant 2\) or \((X) \lt 3\) They get A0 if they write P(\(X \leqslant 2\)/ \(X \lt 3\))
4th A1 \((X) \geqslant 16\) or \((X) \gt 15\) They get A0 if they write P(\(X \geqslant 16\) \(X \gt 15\)
NB these are B1 B1 but mark as A1 A1
\(16 \leqslant X \leqslant 2\) etc is accepted
To describe the critical regions they can use any letter or no letter at all. It does not have to be \(X\).
| Scheme | Marks |
|---|---|
| Actual significance level 0.0021+0.0064=0.0085 or 0.85% | B1 |
| (1) |
Notes
B1 correct answer only
| Scheme | Marks |
|---|---|
| 15 (it) is not in the critical region not significant No significant evidence of a change in \(p = 0.3\) accept H0, (reject H1) \(\mathrm{P}(x \geqslant 15) = 0.0169\) | Bft 2, 1, 0 |
| (2) | |
| (10 marks) |
Notes
Follow through 15 and their critical region
B1 for any one of the 5 correct statements up to a maximum of B2
– B1 for any incorrect statements