| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2012 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Paired sample t-test |
| Difficulty | Standard +0.3 This is a straightforward paired t-test application with clear data and standard procedure. Students must state hypotheses, calculate differences, find mean and standard deviation of differences, compute the test statistic, and compare to critical value. While it requires multiple computational steps (9 marks total), it's a textbook example of a routine S4 procedure with no conceptual challenges or novel elements—slightly easier than average due to its mechanical nature. |
| Spec | 5.05b Unbiased estimates: of population mean and variance5.05c Hypothesis test: normal distribution for population mean |
| 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 |
| Answer | Marks |
|---|---|
| \(H_0: \mu_d = 0\); \(H_1: \mu_d > 0\) (or \(H_1: \mu_d < 0\)) 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 marks total) |
| Answer | Marks |
|---|---|
| \(d: 4, -1, 6, 6, 3, -2, 9, -1, 4, 7, -11, 7\) | M1 |
| \(\Sigma d = 31\), \(\Sigma d^2 = 419\); \(\bar{d} = \pm 2.5833\); sd \(= 5.55073\) (or Var \(= 30.8106\)) | A1; A1 |
| \(t = \frac{\pm 2.5833/\sqrt{12}}{5.55073} = \pm 1.612...\) 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 marks total) |
**Part (a)**
| $H_0: \mu_d = 0$; $H_1: \mu_d > 0$ (or $H_1: \mu_d < 0$) 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 marks total) |
**Guidance notes:**
- B1 both hypotheses required
- B1 must be differences
**Part (b)**
| $d: 4, -1, 6, 6, 3, -2, 9, -1, 4, 7, -11, 7$ | M1 |
| $\Sigma d = 31$, $\Sigma d^2 = 419$; $\bar{d} = \pm 2.5833$; sd $= 5.55073$ (or Var $= 30.8106$) | A1; A1 |
| $t = \frac{\pm 2.5833/\sqrt{12}}{5.55073} = \pm 1.612...$ 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 marks total) |
**Guidance 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 $\frac{\pm \text{their } \bar{d} \sqrt{12}}{\text{their sd}}$
- A1 awrt 1.61
- B1 CV
- A1ft follow through their $t$ value – need context of blood pressure and sitting and standing
---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.
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Person & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ & $I$ & $J$ & $K$ & $L$ \\
\hline
Sitting down & 135 & 146 & 138 & 146 & 141 & 158 & 136 & 135 & 146 & 161 & 119 & 151 \\
\hline
Standing up & 131 & 147 & 132 & 140 & 138 & 160 & 127 & 136 & 142 & 154 & 130 & 144 \\
\hline
\end{tabular}
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.
\begin{enumerate}[label=(\alph*)]
\item State clearly the hypotheses that should be used and any necessary assumption that needs to be made. [2]
\item Carry out the test at the 1\% level of significance. [7]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 2012 Q1 [9]}}