| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2020 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Single sample confidence interval t-distribution |
| Difficulty | Standard +0.3 This is a straightforward application of a one-sample t-test with given data. Students must calculate sample mean and standard deviation, perform a standard t-test for H₀: μ = 22 vs H₁: μ > 22, and construct a confidence interval using the t-distribution. While it requires careful calculation and knowledge of the t-test procedure, it's a routine textbook exercise with no conceptual challenges or novel problem-solving required, making it slightly easier than average. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\sum x = 183.6\), \(\sum x^2 = 4249.08\), \(\bar{x} = 22.95\) | B1 | |
| \(s^2 = \frac{1}{7}\!\left(4249.08 - \frac{183.6^2}{8}\right) = 5.066\) | M1 | |
| \(H_0: \mu = 22.0\), \(H_1: \mu > 22.0\) | B1 | |
| \(t = \dfrac{22.95 - 22.0}{\sqrt{\frac{s^2}{8}}} = 1.194\) | M1A1 | |
| Compare \(t\) with correct tabular value 1.895 | M1 | |
| Accept \(H_0\): mean distance thrown is not more than 22.0 m | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(22.95 \pm t\sqrt{\dfrac{s^2}{8}}\) | M1 | |
| With \(t = 2.365\) | B1 | |
| \([21.1,\ 24.8]\) | A1 |
## Question 5:
### Part 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\sum x = 183.6$, $\sum x^2 = 4249.08$, $\bar{x} = 22.95$ | B1 | |
| $s^2 = \frac{1}{7}\!\left(4249.08 - \frac{183.6^2}{8}\right) = 5.066$ | M1 | |
| $H_0: \mu = 22.0$, $H_1: \mu > 22.0$ | B1 | |
| $t = \dfrac{22.95 - 22.0}{\sqrt{\frac{s^2}{8}}} = 1.194$ | M1A1 | |
| Compare $t$ with correct tabular value 1.895 | M1 | |
| Accept $H_0$: mean distance thrown is not more than 22.0 m | A1 | |
### Part 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $22.95 \pm t\sqrt{\dfrac{s^2}{8}}$ | M1 | |
| With $t = 2.365$ | B1 | |
| $[21.1,\ 24.8]$ | A1 | |
---5 A large number of children are competing in a throwing competition. The distances, in metres, thrown by a random sample of 8 children are as follows.\\
$\begin{array} { l l l l l l l l } 19.8 & 22.1 & 24.4 & 21.5 & 20.8 & 26.3 & 23.7 & 25.0 \end{array}$
\begin{enumerate}[label=(\alph*)]
\item Assuming that distances are normally distributed, test, at the $5 \%$ significance level, whether the population mean distance thrown is more than 22.0 metres.
\item Find a 95\% confidence interval for the population mean distance thrown.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2020 Q5 [10]}}