S2 January 2012 Q2
2. David claims that the weather forecasts produced by local radio are no better than those achieved by tossing a fair coin and predicting rain if a head is obtained or no rain if a tail is obtained. He records the weather for 30 randomly selected days. The local radio forecast is correct on 21 of these days.
Test David’s claim at the 5% level of significance.
State your hypotheses clearly. (7)
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : p = 0.5\) | B1 |
| \(\mathrm{H}_1 : p \gt 0.5\) | B1 |
| \(X \sim \mathrm{B}(30, 0.5)\) Using correct Bin | M1 |
| \(\mathrm{P}(X \geqslant 21) = 1 - \mathrm{P}(X \leqslant 20)\) or \(\mathrm{P}(X \leqslant 19) = 0.9506\) \(\mathrm{P}(X \geqslant 20) = 0.0494\) | M1 |
| \(= 1 - 0.9786\) \(= 0.0214\) CR \(X \geqslant 20\) | A1 |
| so significant/reject \(\mathrm{H}_0\) /in Critical region | M1 dep |
| Evidence to suggest David’s claim is incorrect or The weather forecast produced by the local radio is better than those achieved by tossing/flipping a coin | A1 |
| (7) | |
| (7 marks) |
Notes
1st B1 for \(\mathrm{H}_0 : p = 0.5\)
2nd B1 for \(\mathrm{H}_1 : p \gt 0.5\)
SC If both hypotheses are correct but a different letter to \(p\) is used they get B1 B0. If no letter is used they get B0 B0.
1st M1 writing or using B(30,0.5)
One tail
2nd M1 for writing or using \(1 - \mathrm{P}(X \leqslant 20)\) or writing \(\mathrm{P}(X \leqslant 19) = 0.9506\) or \(\mathrm{P}(X \geqslant 20) = 0.0494\). May be implied by correct CR.or probability = 0.0214
A1 for 0.0214 or CR \(X \geqslant 20\)/ \(X \gt 19\). NB \(\mathrm{P}(X \leqslant 20) = 0.9786\) on its own scores M1A1
3rd M1 dependent on the 2nd M1 being awarded. For a correct statement based on the table below. Do not allow non-contextual conflicting statements eg “significant” and “accept H0”. Ignore comparisons.
2nd A1 for a correct contextualised statement. NB A correct contextual statement on its own scores M1A1.
| \(0.05 \lt p \lt 0.95\) | \(p \lt 0.05\) or \(p \gt 0.95\) | |
|---|---|---|
| 3rd M1 | not significant/ accept H0/ Not in CR | significant/ reject H0/ In CR |
| 2nd A1 | David's claim is correct weather forecast produced by the local radio is no better than those achieved by tossing/flipping a coin | David's claim incorrect weather forecast produced by the local radio is better than those achieved by tossing/flipping a coin |
Two tail
1st M1 for writing or using \(1 - \mathrm{P}(X \leqslant 20)\) or writing \(\mathrm{P}(X \leqslant 20) = 0.9786\) or \(\mathrm{P}(X \geqslant 21) = 0.0214\). May be implied by correct CR. or probability = 0.0214 (corrected from the printed mark scheme, which has “probability = 0.197” here)
A1 for 0.0214 or CR \(X \geqslant 21\)/ \(X \gt 20\). NB \(\mathrm{P}(X \leqslant 20) = 0.9786\) on its own scores M1A1
3rd M1 dependent on the 2nd M1 being awarded . For a correct statement based on the table below. Do not allow non-contextual conflicting statements eg“significant” and “accept H0” . Ignore comparisons.
2nd A1 for a correct contextualised statement. NB A correct contextual statement on its own scores M1A1.
| \(0.025 \lt p \lt 0.975\) | \(p \lt 0.025\) or \(p \gt 0.975\) | |
|---|---|---|
| 3rd M1 | not significant/ accept H0/ Not in CR | significant/ reject H0/ In CR |
| 2nd A1 | David's .claim is correct weather forecast produced by the local radio is no better than those achieved by tossing/flipping a coin | David's claim incorrect weather forecast produced by the local radio is better than those achieved by tossing/flipping a coin |