| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Statistics (Further Paper 3 Statistics) |
| Year | 2020 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Single sample t-test |
| Difficulty | Standard +0.3 This is a straightforward one-sample t-test with standard bookwork parts: (a) requires routine application of the t-test procedure with given values, (b) asks for a standard definition of Type II error in context, and (c) involves basic interpretation of a histogram for normality. All parts are textbook exercises requiring recall and direct application rather than problem-solving or insight, making it slightly easier than average. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: \mu = 1.8\), \(H_1: \mu > 1.8\) | B1 | States both hypotheses using correct language |
| \(t = \frac{2 - 1.8}{\frac{0.4}{\sqrt{12}}}\) | M1 | Obtains \(t\) test statistic with 'their' sample mean and variance; condone \(z =\) |
| \(t = 1.73\); \(p = \) AWRT 0.0556 | A1 | Obtains \(t\) statistic correctly; condone \(z =\) |
| \(t_{11}\) at \(95\% = 1.796\); \(1.73 < 1.796\); Accept \(H_0\) | M1 | Evaluates \(t\) model by comparing 'their' test statistic and correct critical value or by comparing \(p\)-value with 0.05 |
| No significant evidence to suggest that the rainfall in February in the town has increased | E1F | Infers \(H_0\) not rejected; FT 'their' comparison using \(t\) model |
| (Contextual conclusion, not definite) | E1F | Concludes in context; FT incorrect rejection of \(H_0\) or 'their' test statistic/p-value/'their' critical value if not |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| To conclude that the rainfall per day in February has not increased when it has | E1 | States accepting mean rainfall per day in February has not increased when it has |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| The assumption of a normal distribution is not appropriate because the distribution is not symmetrical | E1 | Recognises assumption not appropriate because distribution is not symmetrical or is skewed |
## Question 7:
### Part 7(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: \mu = 1.8$, $H_1: \mu > 1.8$ | B1 | States both hypotheses using correct language |
| $t = \frac{2 - 1.8}{\frac{0.4}{\sqrt{12}}}$ | M1 | Obtains $t$ test statistic with 'their' sample mean and variance; condone $z =$ |
| $t = 1.73$; $p = $ AWRT 0.0556 | A1 | Obtains $t$ statistic correctly; condone $z =$ |
| $t_{11}$ at $95\% = 1.796$; $1.73 < 1.796$; Accept $H_0$ | M1 | Evaluates $t$ model by comparing 'their' test statistic and correct critical value or by comparing $p$-value with 0.05 |
| No significant evidence to suggest that the rainfall in February in the town has increased | E1F | Infers $H_0$ not rejected; FT 'their' comparison using $t$ model |
| (Contextual conclusion, not definite) | E1F | Concludes in context; FT incorrect rejection of $H_0$ or 'their' test statistic/p-value/'their' critical value if not |
### Part 7(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| To conclude that the rainfall per day in February has not increased when it has | E1 | States accepting mean rainfall per day in February has not increased when it has |
### Part 7(c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| The assumption of a normal distribution is not appropriate because the distribution is not symmetrical | E1 | Recognises assumption not appropriate because distribution is not symmetrical or is skewed |7 The rainfall per day in February in a particular town has been recorded as having a mean of 1.8 inches.
Sienna claims that rainfall in February has increased in the town. She records the rainfall in a random sample of 12 days.
Her sample mean is 2 inches and her sample standard deviation is 0.4 inches.\\
It is assumed that rainfall per day has a normal distribution.\\
7
\begin{enumerate}[label=(\alph*)]
\item Investigate Sienna's claim using the $5 \%$ level of significance.\\
7
\item For the test carried out in part (a), state in context the meaning of a Type II error.
7
\item The distribution of rainfall per day in February in the town over 10 years is shown in the histogram.\\
\includegraphics[max width=\textwidth, alt={}, center]{443e7f17-a555-41ff-9d91-541cf45aae99-11_508_645_849_699}
Explain whether or not the assumption that rainfall per day in February has a normal distribution is appropriate.
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Statistics 2020 Q7 [8]}}