S2 June 2009 Q2
2. An effect of a certain disease is that a small number of the red blood cells are deformed. Emily has this disease and the deformed blood cells occur randomly at a rate of 2.5 per ml of her blood. Following a course of treatment, a random sample of 2 ml of Emily’s blood is found to contain only 1 deformed red blood cell.
Stating your hypotheses clearly and using a 5% level of significance, test whether or not there has been a decrease in the number of deformed red blood cells in Emily’s blood. (6)
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \lambda = 2.5\ \ (\text{or } \lambda = 5) \qquad \mathrm{H}1 : \lambda \lt 2.5\ \ (\text{or } \lambda \lt 5)\) \(\lambda\) or \(\mu\) | B1B1 |
| \(X \sim \mathrm{Po}(5)\) | M1 |
| \(\mathrm{P}(X \leqslant 1) = 0.0404\) or CR \(X \leqslant 1\) | A1 |
| [0.0404<0.05 ] this is significant or reject \(\mathrm{H}_0\) or it is in the critical region | M1 |
| There is evidence of a decrease in the (mean) number/rate of deformed blood cells | A1 |
| (6) | |
| (6 marks) |
Notes
1st B1 for \(\mathrm{H}_0\) must use lambda or mu; 5 or 2.5.
2nd B1 for \(\mathrm{H}_1\) must use lambda or mu; 5 or 2.5
1st M1 for use of Po(5) may be implied by probability( must be used not just seen) eg. \(\mathrm{P}(X = 1) = 0.0404 - \ldots\) would score M1 A0
1st A1 for 0.0404 seen or correct CR
2nd M1 for a correct statement (this may be contextual) comparing their probability and 0.05 (or comparing 1 with their critical region). Do not allow conflicting statements.
2nd A1 is not a follow through. Need the word decrease, number or rate and deformed blood cells for contextual mark.
If they have used \(\ne\) in \(\mathrm{H}_1\) they could get B1 B0 M1 A1 M1A0
mark as above except they gain the
1st A1 for \(\mathrm{P}(X \leqslant 1) = 0.0404\) or CR \(X \leqslant 0\)
2nd M1 for a correct statement (this may be contextual) comparing their probability and 0.025 (or comparing 1 with their critical region)
They may compare with 0.95 (one tail method) or 0.975 (two tail method)
Probability is 0.9596.
(corrected from the printed mark scheme: printed “0.975 (one tail method)”)