| Exam Board | Edexcel |
|---|---|
| Module | Paper 3 (Paper 3) |
| Session | Specimen |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Unbiased estimates calculation |
| Difficulty | Standard +0.3 This is a standard multi-part statistics question covering routine A-level techniques: histogram calculations, linear interpolation for median, mean/SD from grouped data, and a one-sample z-test. All parts follow textbook procedures with no novel insight required. The hypothesis test is straightforward (given σ, large sample). Slightly easier than average due to predictable structure and provided summary statistics. |
| Spec | 2.02b Histogram: area represents frequency2.02f Measures of average and spread2.02g Calculate mean and standard deviation2.05e Hypothesis test for normal mean: known variance |
| Mass \(w ( \mathrm {~g} )\) | Midpoint \(y ( \mathrm {~g} )\) | Frequency f |
| \(240 \leq w < 245\) | 242.5 | 8 |
| \(245 \leq w < 248\) | 246.5 | 15 |
| \(248 \leq w < 252\) | 250.0 | 35 |
| \(252 \leq w < 255\) | 253.5 | 23 |
| \(255 \leq w < 260\) | 257.5 | 9 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\mathbf{r} = (-4.5\mathbf{i} + 3\mathbf{j})\) | B1 | Correct displacement vector |
| Use of \(\mathbf{r} = \mathbf{u}t + \frac{1}{2}\mathbf{a}t^2\) | M1 | Use of correct strategy and/or formula to give equation in u only (could be obtained by two integrations) |
| \((-4.5\mathbf{i} + 3\mathbf{j}) = 3\mathbf{u} + 0.5(\mathbf{i} - 2\mathbf{j})3^2\) | A1ft | Correct equation in u only, following their displacement vector |
| \(\mathbf{u} = (-3\mathbf{i} + 4\mathbf{j})\) | A1 | Correct answer |
## Question 1:
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\mathbf{r} = (-4.5\mathbf{i} + 3\mathbf{j})$ | B1 | Correct displacement vector |
| Use of $\mathbf{r} = \mathbf{u}t + \frac{1}{2}\mathbf{a}t^2$ | M1 | Use of correct strategy and/or formula to give equation in **u** only (could be obtained by two integrations) |
| $(-4.5\mathbf{i} + 3\mathbf{j}) = 3\mathbf{u} + 0.5(\mathbf{i} - 2\mathbf{j})3^2$ | A1ft | Correct equation in **u** only, following their displacement vector |
| $\mathbf{u} = (-3\mathbf{i} + 4\mathbf{j})$ | A1 | Correct answer |
**Total: 4 marks**
---\begin{enumerate}
\item Kaff coffee is sold in packets. A seller measures the masses of the contents of a random sample of 90 packets of Kaff coffee from her stock. The results are shown in the table below.
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | }
\hline
Mass $w ( \mathrm {~g} )$ & Midpoint $y ( \mathrm {~g} )$ & Frequency f \\
\hline
$240 \leq w < 245$ & 242.5 & 8 \\
\hline
$245 \leq w < 248$ & 246.5 & 15 \\
\hline
$248 \leq w < 252$ & 250.0 & 35 \\
\hline
$252 \leq w < 255$ & 253.5 & 23 \\
\hline
$255 \leq w < 260$ & 257.5 & 9 \\
\hline
\end{tabular}
\end{center}
$$\text { (You may use } \sum \mathrm { fy } ^ { 2 } = 5644 \text { 171.75) }$$
A histogram is drawn and the class $245 \leq w < 248$ is represented by a rectangle of width 1.2 cm and height 10 cm .\\
(a) Calculate the width and the height of the rectangle representing the class $255 \leq w < 260$.\\
(b) Use linear interpolation to estimate the median mass of the contents of a packet of Kaff coffee to 1 decimal place.\\
(c) Estimate the mean and the standard deviation of the mass of the contents of a packet of Kaff coffee to 1 decimal place.
The seller claims that the mean mass of the contents of the packets is more than the stated mass. Given that the stated mass of the contents of a packet of Kaff coffee is 250 g and the actual standard deviation of the contents of a packet of Kaff coffee is 4 g ,\\
(d) test, using a 5\% level of significance, whether or not the seller's claim is justified. State your hypotheses clearly.\\
(You may assume that the mass of the contents of a packet is normally distributed.)\\
(e) Using your answers to parts (b) and (c), comment on the assumption that the mass of the contents of a packet is normally distributed.\\
(Total 14 marks)\\
\hfill \mbox{\textit{Edexcel Paper 3 Q1 [14]}}