| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2021 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Paired sample t-test |
| Difficulty | Standard +0.8 This is a paired t-test requiring students to test a one-sided hypothesis about a specific difference (μ > 2), not just μ > 0. Students must calculate differences, find mean and standard deviation, formulate correct hypotheses (H₀: μ = 2 vs H₁: μ > 2), compute the test statistic using the shifted null hypothesis, compare with t₉ critical value, and state the normality assumption. The non-zero null hypothesis value adds conceptual complexity beyond standard paired t-test questions, though the calculations themselves are routine for Further Statistics students. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Athlete | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) | \(J\) |
| Before | 150 | 146 | 131 | 135 | 126 | 142 | 130 | 129 | 137 | 134 |
| After | 145 | 138 | 129 | 135 | 122 | 135 | 132 | 128 | 127 | 137 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: \mu_B - \mu_A = 2\) and \(H_1: \mu_B - \mu_A > 2\) | B1 | Or use of \(\mu_d\) |
| Differences: \(5\ 8\ 2\ 0\ 4\ 7\ -2\ 1\ 10\ -3\) | M1 | Allow one error |
| \(\sum d = 32\), \(\sum d^2 = 272\), \(\bar{d} = 3.2\), \(s^2 = \frac{1}{9}\left(272 - \frac{32^2}{10}\right) = 18.84\ \left(=\frac{848}{45}\right)\) | M1 | Sample mean and variance |
| \(t = \frac{3.2 - 2}{\sqrt{\frac{s^2}{10}}} = 0.874\) | M1 A1 | |
| Compare with tabular value \(1.833\): \(0.874 < 1.833\). Accept \(H_0\). | M1 | Compare their value with 1.833 and conclusion |
| Insufficient evidence to support claim. | A1 | Correct conclusion, in context, following correct work. Level of uncertainty in language is used |
| Assumption: population differences are normally distributed | B1 |
## Question 4:
$H_0: \mu_B - \mu_A = 2$ and $H_1: \mu_B - \mu_A > 2$ | B1 | Or use of $\mu_d$ |
Differences: $5\ 8\ 2\ 0\ 4\ 7\ -2\ 1\ 10\ -3$ | M1 | Allow one error |
$\sum d = 32$, $\sum d^2 = 272$, $\bar{d} = 3.2$, $s^2 = \frac{1}{9}\left(272 - \frac{32^2}{10}\right) = 18.84\ \left(=\frac{848}{45}\right)$ | M1 | Sample mean and variance |
$t = \frac{3.2 - 2}{\sqrt{\frac{s^2}{10}}} = 0.874$ | M1 A1 | |
Compare with tabular value $1.833$: $0.874 < 1.833$. Accept $H_0$. | M1 | Compare their value with 1.833 and conclusion |
Insufficient evidence to support claim. | A1 | Correct conclusion, in context, following correct work. Level of uncertainty in language is used |
Assumption: population differences are normally distributed | B1 | |4 Manet has developed a new training course to help athletes improve their time taken to run 800 m . Manet claims that his course will decrease an athlete's time by more than 2 s on average. For a random sample of 10 athletes the times taken, in seconds, before and after the course are given in the following table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | }
\hline
Athlete & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ & $I$ & $J$ \\
\hline
Before & 150 & 146 & 131 & 135 & 126 & 142 & 130 & 129 & 137 & 134 \\
\hline
After & 145 & 138 & 129 & 135 & 122 & 135 & 132 & 128 & 127 & 137 \\
\hline
\end{tabular}
\end{center}
Use a $t$-test, at the $5 \%$ significance level, to test whether Manet's claim is justified, stating any assumption that you make.\\
\hfill \mbox{\textit{CAIE Further Paper 4 2021 Q4 [8]}}