| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2023 |
| Session | November |
| Marks | 8 |
| 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 with clear data and standard procedure. Students calculate differences, find mean and standard deviation, then apply the t-test formula with a one-tailed hypothesis at 2.5% level. The only minor complication is testing for a reduction 'by more than 2 units' rather than just a reduction, requiring μ_d > 2 in the null hypothesis setup. This is slightly easier than average as it's a textbook application with no conceptual tricks. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.05c Hypothesis test: normal distribution for population mean |
| \cline { 2 - 12 } \multicolumn{1}{c|}{} | Person | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) | |||
\multirow{2}{*}{
| Beginning | 72 | 84 | 120 | 90 | 102 | 135 | 64 | 75 | 80 | 88 | ||
| \cline { 2 - 12 } | End | 64 | 76 | 105 | 92 | 105 | 115 | 67 | 75 | 75 | 84 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0: \mu_B - \mu_E = 2\); \(H_1: \mu_B - \mu_E > 2\) | B1 | May use \(\mu_d\), but must be consistent with direction of differences found |
| Differences: 8, 8, 15, \(-2\), \(-3\), 20, \(-3\), 0, 5, 4 | M1 | At most 2 errors |
| \(\sum d = 52\), \(\sum d^2 = 816\), \(s^2 = \frac{1}{9}\left(816 - \frac{52^2}{10}\right) = \frac{2728}{45} = 60.62\) | M1 | *Their* values but must see 9 and 10 used correctly |
| \(t = \dfrac{\frac{52}{10} - 2}{\sqrt{\frac{s^2}{10}}}\) | M1 | |
| \(t = 1.29967\), \(1.30\) | A1 | |
| Tabular value \(= 2.262\): \(1.30 < 2.262\), accept \(H_0\) | M1 | Allow 'not significant' |
| Insufficient evidence to suggest that cholesterol level has reduced by more than 2 | A1 | CWO. Correct conclusion in context, level of uncertainty in language. A0 if hypotheses wrong way round or missing |
| Total: 7 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Population differences are normally distributed | B1 | |
| Total: 1 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: \mu_B - \mu_E = 2$; $H_1: \mu_B - \mu_E > 2$ | B1 | May use $\mu_d$, but must be consistent with direction of differences found |
| Differences: 8, 8, 15, $-2$, $-3$, 20, $-3$, 0, 5, 4 | M1 | At most 2 errors |
| $\sum d = 52$, $\sum d^2 = 816$, $s^2 = \frac{1}{9}\left(816 - \frac{52^2}{10}\right) = \frac{2728}{45} = 60.62$ | M1 | *Their* values but must see 9 and 10 used correctly |
| $t = \dfrac{\frac{52}{10} - 2}{\sqrt{\frac{s^2}{10}}}$ | M1 | |
| $t = 1.29967$, $1.30$ | A1 | |
| Tabular value $= 2.262$: $1.30 < 2.262$, accept $H_0$ | M1 | Allow 'not significant' |
| Insufficient evidence to suggest that cholesterol level has reduced by more than 2 | A1 | CWO. Correct conclusion in context, level of uncertainty in language. A0 if hypotheses wrong way round or missing |
| **Total: 7** | | |
---
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| **Population differences** are **normally** distributed | B1 | |
| **Total: 1** | | |
---3 Scientists are studying the effects of exercise on LDL blood cholesterol levels. Over a three-month period, a large group of people exercised for 20 minutes each day. For a randomly chosen sample of 10 of these people, the LDL blood cholesterol levels were measured at the beginning and the end of the three-month period. The results, measured in suitable units, are as follows.
\begin{center}
\begin{tabular}{ | c | l | c | c | c | c | c | c | c | c | c | c | }
\cline { 2 - 12 }
\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & Person & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ & $I$ \\
\hline
\multirow{2}{*}{\begin{tabular}{ c }
Cholesterol \\
level \\
\end{tabular}} & Beginning & 72 & 84 & 120 & 90 & 102 & 135 & 64 & 75 & 80 & 88 \\
\cline { 2 - 12 }
& End & 64 & 76 & 105 & 92 & 105 & 115 & 67 & 75 & 75 & 84 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Test, at the $2.5 \%$ significance level, whether there is evidence that the population mean LDL blood cholesterol level has reduced by more than 2 units after the three-month period.
\item State any assumption that you have made in part (a).\\
\includegraphics[max width=\textwidth, alt={}, center]{b6635fbc-3c9d-4f93-b51a-b1cbd71ddbb1-06_399_1383_269_324}
As shown in the diagram, the continuous random variable $X$ has probability density function f given by
$$f ( x ) = \begin{cases} m x & 0 \leqslant x \leqslant 2 \\ \frac { k } { x ^ { 2 } } + c & 2 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$
where $m , k$ and $c$ are constants.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2023 Q3 [8]}}