S2 January 2008 Q5
5. Dhriti grows tomatoes. Over a period of time, she has found that there is a probability 0.3 of a ripe tomato having a diameter greater than 4 cm. She decides to try a new fertiliser. In a random sample of 40 ripe tomatoes, 18 have a diameter greater than 4 cm. Dhriti claims that the new fertiliser has increased the probability of a ripe tomato being greater than 4 cm in diameter.
Test Dhriti’s claim at the 5% level of significance. State your hypotheses clearly. (7)
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : p = 0.3;\quad \mathrm{H}_1 : p \gt 0.3\) | B1 B1 |
| Let X represent the number of tomatoes greater than 4 cm : \(X \sim \mathrm{B}(40, 0.3)\) | B1 |
| \(\mathrm{P}(X \geqslant 18) = 1 - \mathrm{P}(X \leqslant 17)\) \(\mathrm{P}(X \geqslant 18)\ 1 - \mathrm{P}(X \leqslant 17) = 0.0320\) \(\mathrm{P}(X \geqslant 17) = 1 - \mathrm{P}(X \leqslant 16) = 0.0633\) | M1 |
| \(= 0.0320\) CR \(X \geqslant 18\) | A1 |
| \(0.0320 \lt 0.05\) \(18 \geqslant 18\) or 18 in the critical region Reject \(\mathrm{H}_0\) or it is significant | M1 |
| New fertiliser has increased the probability of a tomato being greater than 4 cm Or Dhriti’s claim is true | B1d cao |
| (7) | |
| (7 marks) |
Notes
B1 for correct \(\mathrm{H}_0\). must use \(p\) or pi
B1 for correct \(\mathrm{H}_1\) must use \(p\) and be one tail.
B1 using B(40, 0.3). This may be implied by their calculation
M1 attempt to find \(1 - \mathrm{P}(X \leqslant 17)\) or get a correct probability.
For CR method must attempt to find \(\mathrm{P}(X \geqslant 18)\) or give the correct critical region
A1 awrt 0.032 or correct CR.
M1 correct statement based on their probability, \(\mathrm{H}_1\) and 0.05
or a correct contextualised statement that implies that.
B1 this is not a follow through. conclusion in context. Must use the words increased, tomato and some reference to size or diameter. This is dependent on them getting the previous M1
If they do a two tail test they may get
B1 B0 B1 M1 A1 M1 B0
For the second M1 they must have accept Ho or it is not significant
or a correct contextualised statement that implies that.
(corrected from the printed mark scheme: printed “no evidence to Reject \(\mathrm{H}_0\) or it is significant”; since \(0.0320 \lt 0.05\) the result is significant and \(\mathrm{H}_0\) is rejected)