S2 January 2006 Q7
7. A teacher thinks that 20% of the pupils in a school read the Deano comic regularly.
He chooses 20 pupils at random and finds 9 of them read the Deano.
(a)
(i) Test, at the 5% level of significance, whether or not there is evidence that the percentage of pupils that read the Deano is different from 20%. State your hypotheses clearly.
(ii) State all the possible numbers of pupils that read the Deano from a sample of size 20 that will make the test in part (a)(i) significant at the 5% level. (9)
The teacher takes another 4 random samples of size 20 and they contain 1, 3, 1 and 4 pupils that read the Deano.
(b) By combining all 5 samples and using a suitable approximation test, at the 5% level of significance, whether or not this provides evidence that the percentage of pupils in the school that read the Deano is different from 20%. (8)
(c) Comment on your results for the tests in part (a) and part (b). (2)
| Scheme | Marks |
|---|---|
| Two tail (i) \(\mathrm{H}_0: p = 0.2,\ \mathrm{H}_1: p \neq 0.2\) | B1B1 |
| \(\mathrm{P}(X \geqslant 9) = 1 - \mathrm{P}(X \leqslant 8) \quad\) or attempt critical value/region \(= 1 - 0.9900 = 0.01 \qquad \text{CR } X \geqslant 9\) | M1 |
| \(0.01 \lt 0.025\) or \(9 \geqslant 9\) or \(0.99 \gt 0.975\) or \(0.02 \lt 0.05\) or lies in interval with correct interval stated. | A1 |
| Evidence that the percentage of pupils that read Deano is not 20% | A1 |
| (ii) \(X \sim \mathrm{Bin}(20, 0.2)\) | B1 |
| So 0 or [9,20] make test significant. | B1B1B1 |
| (9) |
Notes
B1B1 \(p =\)
B1 (ii) may be implied or seen in (i) or (ii)
B1B1B1 0, 9, between “their 9” and 20
One tail
| Scheme | Marks |
|---|---|
| (i) \(\mathrm{H}_0: p = 0.2,\ \mathrm{H}_1: p \gt 0.2\) | B1B0 |
| \(\mathrm{P}(X \geqslant 9) = 1 - \mathrm{P}(X \leqslant 8) \quad\) or attempt critical value/region | M1 |
| \(= 1 - 0.9900 = 0.01 \qquad \text{CR } X \geqslant 8\) | A0 |
| \(0.01 \lt 0.05\) or \(9 \geqslant 8\) (therefore Reject \(\mathrm{H}_0\)), evidence that the percentage of pupils that read Deano is not 20% | A1 |
| (ii) \(X \sim \mathrm{Bin}(20, 0.2)\) | B1 |
| So 0 or [8,20] make test significant. | B1B0B1 |
| (9) |
B1 (ii) may be implied or seen in (i) or (ii)
B1B0B1 0, 9, between “their 8” and 20
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0: p = 0.2,\ \mathrm{H}_1: p \neq 0.2\) | B1 |
| \(W \sim \mathrm{Bin}(100, 0.2)\) \(W \sim \mathrm{N}(20, 16)\) | B1; B1 |
| \(\mathrm{P}(X \leqslant 18) = \mathrm{P}\left(Z \leqslant \dfrac{18.5 - 20}{4}\right) \quad\) or \(\dfrac{x\left(+\frac{1}{2}\right) - 20}{4} = \pm 1.96\) \(= \mathrm{P}(Z \leqslant -0.375)\) | M1M1A1 |
| \(= 0.352 - 0.354 \qquad \text{CR } X \lt 12.16\) or 11.66 for \(\frac{1}{2}\) | A1 |
| [\(0.352 \gt 0.025\) or \(18 \gt 12.16\) therefore insufficient evidence to reject \(\mathrm{H}_0\)] Combined numbers of Deano readers suggests 20% of pupils read Deano | A1 |
| (8) |
Notes
B1; B1 normal; 20 and 16
M1M1A1 \(\pm\) cc, standardise or use z value, standardise
One tail
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0: p = 0.2,\ \mathrm{H}_1: p \lt 0.2\) | B1ft |
| \(W \sim \mathrm{Bin}(100, 0.2)\) \(W \sim \mathrm{N}(20, 16)\) | B1; B1 |
| \(\mathrm{P}(X \leqslant 18) = \mathrm{P}\left(Z \leqslant \dfrac{18.5 - 20}{4}\right) \quad\) or \(\dfrac{x - 20}{4} = -1.6449\) \(= \mathrm{P}(Z \leqslant -0.375)\) | M1M1A1 |
| \(= 0.3520 \qquad \text{CR } X \lt 13.4\) or 12.9 | A1 |
| [\(0.352 \gt 0.05\) or \(18 \gt 13.4\) therefore insufficient evidence to reject \(\mathrm{H}_0\)] Combined numbers of Deano readers suggests 20% of pupils read Deano | A1 |
| (8) |
B1; B1 normal; 20 and 16
M1M1A1 \(\pm\) cc, standardise or standardise, use z value
A1 awrt 0.352
| Scheme | Marks |
|---|---|
| Conclusion that they are different. | B1 |
| Either large sample size gives better result Or Looks as though they are not all drawn from the same population. | B1 |
| (2) | |
| (19 marks) |