S4 June 2012 Q1
1. A medical student is investigating whether there is a difference in a person’s blood pressure when sitting down and after standing up. She takes a random sample of 12 people and measures their blood pressure, in mmHg, when sitting down and after standing up.
The results are shown below.
| Person | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) | \(J\) | \(K\) | \(L\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Sitting down | 135 | 146 | 138 | 146 | 141 | 158 | 136 | 135 | 146 | 161 | 119 | 151 |
| Standing up | 131 | 147 | 132 | 140 | 138 | 160 | 127 | 136 | 142 | 154 | 130 | 144 |
The student decides to carry out a paired \(t\)-test to investigate whether, on average, the blood pressure of a person when sitting down is more than their blood pressure after standing up.
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \mu_d = 0,\quad \mathrm{H}_1 : \mu_d \gt 0\) (or \(\mathrm{H}_1 : \mu_d \lt 0\)) where \(\mu_d\) is the (population) mean difference :- BP sitting down – BP standing. (BP standing – BP sitting down) | B1 |
| Assume the differences are normally distributed | B1 |
| (2) |
Notes
B1 both hypotheses.
B1 must be differences
| Scheme | Marks |
|---|---|
| \(d\): 4, −1, 6, 6, 3, −2, 9, −1, 4, 7, −11, 7 | M1 |
| \((\Sigma d = 31,\ \Sigma d^2 = 419)\quad \bar{d} = \pm 2.5833;\ \mathrm{sd} = 5.55073.\ (\text{ or Var} = 30.8106)\) | A1; A1 |
| \(t = \dfrac{\pm 2.5833\sqrt{12}}{5.55073} = \pm 1.612\ldots\) Formula and substitution, 1.61 | M1, A1 |
| Critical value \(t_{11}(1\%) = 2.718\) (1 tail) | B1 |
| Not significant. Insufficient evidence to support that the blood pressure of a person sitting down is more than the blood pressure of a person after standing up. | A1 ft |
| (7) | |
| (9 marks) |
Notes
M1 at least 2 correct or may be implied by correct \(\Sigma d\) or \(\Sigma d^2\) or \(\bar{d}\) or sd or var or implied by correct \(t\) value
A1 correct \(\bar{d}\) awrt \(\pm 2.58\)- may be implied by correct \(t\) value
A1 correct sd awrt 5.55 or var awrt 30.8 - may be implied by correct \(t\) value
M1 \(\dfrac{\pm\text{their }\bar{d}\sqrt{12}}{\text{their sd}}\)
A1 awrt 1.61
B1 CV
A1ft follow through their \(t\) value – need context of blood pressure and sitting and standing