| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2016 |
| 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 standard paired t-test application with clearly paired data, requiring routine calculation of differences, sample mean/standard deviation, and comparison to critical value. While it's S4 (Further Maths), the question follows a textbook template with no conceptual challenges—slightly easier than average due to its mechanical nature, though the Further Maths context prevents it from being significantly negative. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank5.07d Paired vs two-sample: selection |
| Person | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) | \(J\) |
| Weight before diet (kg) | 96 | 110 | 116 | 98 | 121 | 91 | 98 | 106 | 110 | 116 |
| Weight after diet (kg) | 91 | 101 | 111 | 96 | 121 | 91 | 90 | 101 | 104 | 110 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance Notes |
| \(d\): 5, 9, 5, 2, 0, 0, 8, 5, 6, 6 | M1 | for attempting the \(d\)s |
| \(\bar{d} = \frac{\Sigma d}{n} = 4.6\) | M1 | for attempting \(\bar{d}\) |
| \(s^2 = \frac{296 - 10 \times 4.6^2}{9} = 9.378\) | M1 | for \(s_d\) or \(s_d^2\) |
| \(H_0: \mu_d = 2 \quad H_1: \mu_d > 2\) | B1 | for both hypotheses correct in terms of \(\mu\) or \(\mu_d\) (allow a defined symbol) |
| \(t = \pm\frac{4.6 - 2}{\sqrt{\frac{9.378}{10}}} = \pm 2.6848\) | M1 A1 | M1 for attempting correct test statistic \(\frac{\bar{d}}{s_d/\sqrt{10}}\); A1 awrt 2.68 |
| \(t_9(5\%) = \pm 1.833\ldots\) | B1 | awrt 1.83 |
| There is evidence to reject \(H_0\). There is sufficient evidence to support the designers claim. | A1ft | A1ft for a correct comment in context |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance Notes |
| The differences in weights are normally distributed. | B1 | for a comment that mentions "differences" and "normal" distribution |
**Question 1:**
**Part (a):**
| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| $d$: 5, 9, 5, 2, 0, 0, 8, 5, 6, 6 | M1 | for attempting the $d$s |
| $\bar{d} = \frac{\Sigma d}{n} = 4.6$ | M1 | for attempting $\bar{d}$ |
| $s^2 = \frac{296 - 10 \times 4.6^2}{9} = 9.378$ | M1 | for $s_d$ or $s_d^2$ |
| $H_0: \mu_d = 2 \quad H_1: \mu_d > 2$ | B1 | for both hypotheses correct in terms of $\mu$ or $\mu_d$ (allow a defined symbol) |
| $t = \pm\frac{4.6 - 2}{\sqrt{\frac{9.378}{10}}} = \pm 2.6848$ | M1 A1 | M1 for attempting correct test statistic $\frac{\bar{d}}{s_d/\sqrt{10}}$; A1 awrt 2.68 |
| $t_9(5\%) = \pm 1.833\ldots$ | B1 | awrt 1.83 |
| There is evidence to reject $H_0$. There is sufficient evidence to support the designers claim. | A1ft | A1ft for a correct comment in context |
**Part (b):**
| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| The **differences** in weights are **normally** distributed. | B1 | for a comment that mentions "differences" and "normal" distribution |\begin{enumerate}
\item A new diet has been designed. Its designers claim that following the diet for a month will result in a mean weight loss of more than 2 kg . In a trial, a random sample of 10 people followed the new diet for a month. Their weights, in kg, before starting the diet and their weights after following the diet for a month were recorded. The results are given in the table below.
\end{enumerate}
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | }
\hline
Person & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ & $I$ & $J$ \\
\hline
Weight before diet (kg) & 96 & 110 & 116 & 98 & 121 & 91 & 98 & 106 & 110 & 116 \\
\hline
Weight after diet (kg) & 91 & 101 & 111 & 96 & 121 & 91 & 90 & 101 & 104 & 110 \\
\hline
\end{tabular}
\end{center}
(a) Using a suitable $t$-test, at the $5 \%$ level of significance, state whether or not the trial supports the designers' claim. State your hypotheses and show your working clearly.\\
(b) State an assumption necessary for the test in part (a).\\
\hfill \mbox{\textit{Edexcel S4 2016 Q1 [9]}}