S2 January 2011 Q2
2. A student takes a multiple choice test. The test is made up of 10 questions each with 5 possible answers. The student gets 4 questions correct. Her teacher claims she was guessing the answers. Using a one tailed test, at the 5% level of significance, test whether or not there is evidence to reject the teacher’s claim.
State your hypotheses clearly. (6)
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : p = 0.2 \qquad \mathrm{H}_1 : p \gt 0.2\) | B1 |
| Under \(\mathrm{H}_0\), \(X \sim \mathrm{Bin}(10, 0.2)\) | B1 |
| \(\mathrm{P}(X \geqslant 4) = 1 - \mathrm{P}(X \leqslant 3)\) OR \(\mathrm{P}(X \leqslant 4) = 0.9672\) | M1 |
| \(= 1 - 0.8791\) \(\mathrm{P}(X \geqslant 5) = 0.0328\) \(= 0.1209\) CR \(X \geqslant 5\) | A1 |
| 0.1209>0.05. Insufficient evidence to reject \(\mathrm{H}_0\) so teacher’s claim is supported. | M1A1ft |
| (6 marks) |
Notes
B1 for both H0 and H1 correct. Must use \(p\) or \(\pi\) (pi)
B1 for writing or using Bin(10,0.2)
M1 for finding or writing \(1 - \mathrm{P}(X \leqslant 3)\) or \(\mathrm{P}(X \leqslant 4) = 0.9672\)
\(\mathrm{P}(X \geqslant 5) = 0.0328\) oe or a correct critical region
A1 awrt 0.121 or CR \(X \geqslant 5\)
M1 need \(p\)<0.5 and:
correct statement using their Probability and 0.05 if one tail test or
correct statement using their Probability and 0.025 if two tail test (condone a comparison with 0.05 instead of 0.025 for a two tail test).
Do not allow non-contextual conflicting statements eg “significant” and “accept H0”
A1ft correct contextual statement followed through from “their prob”.
Either a comment on whether the teacher’s claim was correct or on whether the student was guessing the answers.
NB if a correct contextual statement only is given for their probability then award M1 A1
If \(p\)>0.5
They may compare with 0.95 (one tail method) or 0.975 (two tail method)
Probability is 0.8791.