| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Paired t-test |
| Difficulty | Standard +0.3 This is a standard paired t-test application with straightforward calculations (differences, mean, standard deviation) and hypothesis testing at given significance levels. The second part requires finding a confidence interval bound, which is routine A-level further maths content. While it involves multiple steps, each is procedural with no novel insight required. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Patient | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Pre-treatment grip | 24.3 | 29.5 | 28.0 | 28.5 | 21.5 | 28.7 | 25.1 | 26.3 |
| Post-treatment grip | 28.3 | 34.6 | 30.3 | 31.6 | 21.5 | 29.8 | 26.0 | 27.5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State valid assumption (A.E.F.): Population of differences has Normal distribution | B1 | |
| State hypotheses: \(H_0: \mu_2 - \mu_1 = 0\), \(H_1: \mu_2 - \mu_1 > 0\) | B1 | |
| Consider differences e.g.: \(4.0\quad 5.1\quad 2.3\quad 3.1\quad 0\quad 1.1\quad 0.9\quad 1.2\) | M1 | |
| Calculate sample mean: \(\bar{x} = 17.7/8\) \([= 2.2125]\) | M1 | |
| Estimate population variance: \(s^2 = (60.37 - 17.7^2/8)/7\) | M1 | |
| (allow biased: \(2.6511\) or \(1.628^2\)) \([= 3.0298\) or \(1.741^2]\) | ||
| Calculate value of \(t\) (to 2 dp): \(t = \bar{x}/(s/\sqrt{8}) = 3.60\) [or \(3.59\)] | M1 *A1 | |
| Compare with correct tabular \(t\) value: \(t_{7,\,0.99} = 2.998\) (to 2 dp) | *B1 | |
| Correct conclusion (AEF, dep *A1, *B1): There is an increase | B1 | |
| Formulate inequality with any tabular \(t\) value: \((\bar{x} - w)/(s/\sqrt{8}) > t_{7,\,0.9}\) | M1 | |
| Use correct tabular value (to 2 dp): \(t_{7,\,0.9} = 1.415\) | A1 | |
| Evaluate inequality for \(w\): \(w < 1.34\) [or \(\leq\)] | A1 | |
| S.R. Allow M1 A1 A0 if \(=\) or \(<\) used in inequality | Total: 12 |
## Question 10:
| Answer/Working | Mark | Guidance |
|---|---|---|
| State valid assumption (A.E.F.): Population of differences has Normal distribution | B1 | |
| State hypotheses: $H_0: \mu_2 - \mu_1 = 0$, $H_1: \mu_2 - \mu_1 > 0$ | B1 | |
| Consider differences e.g.: $4.0\quad 5.1\quad 2.3\quad 3.1\quad 0\quad 1.1\quad 0.9\quad 1.2$ | M1 | |
| Calculate sample mean: $\bar{x} = 17.7/8$ $[= 2.2125]$ | M1 | |
| Estimate population variance: $s^2 = (60.37 - 17.7^2/8)/7$ | M1 | |
| (allow biased: $2.6511$ or $1.628^2$) $[= 3.0298$ or $1.741^2]$ | | |
| Calculate value of $t$ (to 2 dp): $t = \bar{x}/(s/\sqrt{8}) = 3.60$ [or $3.59$] | M1 *A1 | |
| Compare with correct tabular $t$ value: $t_{7,\,0.99} = 2.998$ (to 2 dp) | *B1 | |
| Correct conclusion (AEF, dep *A1, *B1): There is an increase | B1 | |
| Formulate inequality with any tabular $t$ value: $(\bar{x} - w)/(s/\sqrt{8}) > t_{7,\,0.9}$ | M1 | |
| Use correct tabular value (to 2 dp): $t_{7,\,0.9} = 1.415$ | A1 | |
| Evaluate inequality for $w$: $w < 1.34$ [or $\leq$] | A1 | |
| **S.R.** Allow M1 A1 A0 if $=$ or $<$ used in inequality | | **Total: 12** |
---10 Carpal Tunnel syndrome is a condition which affects a person's ability to grip with their hands. Researchers tested a treatment for this syndrome which was applied to 8 randomly chosen patients. A pre-treatment and a post-treatment test of grip was given to each patient, with the following results, measured in kg.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | }
\hline
Patient & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
Pre-treatment grip & 24.3 & 29.5 & 28.0 & 28.5 & 21.5 & 28.7 & 25.1 & 26.3 \\
\hline
Post-treatment grip & 28.3 & 34.6 & 30.3 & 31.6 & 21.5 & 29.8 & 26.0 & 27.5 \\
\hline
\end{tabular}
\end{center}
Stating any required assumption, test, at the $1 \%$ significance level, whether the mean grip of people with the syndrome increases after undergoing the treatment.
It is given that there is evidence at the $10 \%$ significance level that the mean grip increases by more than $w \mathrm {~kg}$. Find an inequality for $w$.
\hfill \mbox{\textit{CAIE FP2 2010 Q10 [12]}}