S2 June 2007 Q2
2. Bacteria are randomly distributed in a river at a rate of 5 per litre of water. A new factory opens and a scientist claims it is polluting the river with bacteria. He takes a sample of 0.5 litres of water from the river near the factory and finds that it contains 7 bacteria. Stating your hypotheses clearly test, at the 5% level of significance, the claim of the scientist. (7)
One tail test Method 1
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \lambda = 5 \ (\lambda = 2.5)\) | B1 |
| \(\mathrm{H}_1 : \lambda \gt 5 \ (\lambda \gt 2.5)\) | B1 |
| \(X \sim \mathrm{Po}(2.5)\) | M1 |
| \(\mathrm{P}(X \geqslant 7) = 1 - \mathrm{P}(X \leqslant 6)\) \(\left[\mathrm{P}(X \geqslant 5) = 1 - 0.8912 = 0.1088\right.\) \(\qquad\qquad\ \ = 1 - 0.9858\) \(\left.\mathrm{P}(X \geqslant 6) = 1 - 0.9580 = 0.0420\right]\) | M1 |
| \(= 0.0142\) CR \(X \geqslant 6\) | A1 |
| \(0.0142 \lt 0.05\) \(7 \geqslant 6\) or 7 is in critical region or 7 is significant | M1 |
| (Reject \(\mathrm{H}_0\).) There is significant evidence at the 5% significance level that the factory is polluting the river with bacteria. or The scientists claim is justified | B1 |
| (7) | |
| (7 marks) |
Notes
1st B1 may use \(\lambda\) or \(\mu\)
1st M1 may be implied
2nd M1 att \(\mathrm{P}(X \geqslant 7)\) | \(\mathrm{P}(X \geqslant 6)\)
A1 awrt 0.0142
One tail test: Method 2
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \lambda = 5 \ (\lambda = 2.5)\) | B1 |
| \(\mathrm{H}_1 : \lambda \gt 5 \ (\lambda \gt 2.5)\) | B1 |
| \(X \sim \mathrm{Po}(2.5)\) | M1 |
| \(\mathrm{P}(X \lt 7)\) \(\left[\mathrm{P}(X \lt 5) = 0.8912\right]\), \(\mathrm{P}(X \lt 6) = 0.9580\) \(= 0.9858\) CR \(X \geqslant 6\) | M1 A1 |
| \(0.9858 \gt 0.95\) \(7 \geqslant 6\) or 7 is in critical region or 7 is significant | M1 |
| (Reject \(\mathrm{H}_0\).) There is significant evidence at the 5% significance level that the factory is polluting the river with bacteria. or The scientists claim is justified | B1 |
| (7) |
1st B1 may use \(\lambda\) or \(\mu\)
1st M1 may be implied
2nd M1 att \(\mathrm{P}(X \lt 7)\) | \(\mathrm{P}(X \lt 6)\)
A1 wrt 0.986
Two tail test: Method 1
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \lambda = 5 \ (\lambda = 2.5)\) | B1 |
| \(\mathrm{H}_1 : \lambda \ne 5 \ (\lambda \ne 2.5)\) | B0 |
| \(X \sim \mathrm{Po}(2.5)\) | M1 |
| \(\mathrm{P}(X \geqslant 7) = 1 - \mathrm{P}(X \leqslant 6)\) \(\left[\mathrm{P}(X \geqslant 6) = 1 - 0.9580 = 0.0420\right.\) \(\qquad\qquad\ \ = 1 - 0.9858\) \(\left.\mathrm{P}(X \geqslant 7) = 1 - 0.9858 = 0.0142\right]\) | M1 |
| \(= 0.0142\) CR \(X \geqslant 7\) | A1 |
| \(0.0142 \lt 0.025\) \(7 \geqslant 7\) or 7 is in critical region or 7 is significant | M1 |
| (Reject \(\mathrm{H}_0\).) There is significant evidence at the 5% significance level that the factory is polluting the river with bacteria. or The scientists claim is justified | B1 |
| (7) |
1st B1 may use \(\lambda\) or \(\mu\)
2nd M1 att \(\mathrm{P}(X \geqslant 7)\) | \(\mathrm{P}(X \geqslant 7)\)
A1 awrt 0.0142
Two tail test: Method 2
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \lambda = 5 \ (\lambda = 2.5)\) | B1 |
| \(\mathrm{H}_1 : \lambda \ne 5 \ (\lambda \ne 2.5)\) | B0 |
| \(X \sim \mathrm{Po}(2.5)\) | M1 |
| \(\mathrm{P}(X \lt 7)\) \(\left[\mathrm{P}(X \lt 6) = 0.9580\right]\), \(\mathrm{P}(X \lt 7) = 0.9858\) \(= 0.9858\) CR \(X \geqslant 7\) | M1 A1 |
| \(0.9858 \gt 0.975\) \(7 \geqslant 7\) or 7 is in critical region or 7 is significant | M1 |
| (Reject \(\mathrm{H}_0\).) There is significant evidence at the 5% significance level that the factory is polluting the river with bacteria. or The scientists claim is justified | B1 |
| (7) |
1st B1 may use \(\lambda\) or \(\mu\)
2nd M1 att \(\mathrm{P}(X \lt 7)\) | \(\mathrm{P}(X \lt 7)\)
A1 awrt 0.986