| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Statistics (Further Paper 3 Statistics) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Estimated variance confidence interval |
| Difficulty | Standard +0.3 This is a straightforward confidence interval and hypothesis test question using the t-distribution with given summary statistics. Part (a) requires standard calculation of sample mean, variance, and CI using t-tables; part (b) simply checks if the hypothesized value lies in the CI; part (c) asks for basic interpretation. All steps are routine applications of formulas with no conceptual challenges or novel problem-solving required. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\bar{x} = 86.5\), \(s^2 = 28.45\) | B1 (AO 1.1b) | \(s^2\) AWRT \(28.45\) or \(28.455\) (3 d.p.); or \(s\) AWRT \(5.334\); PI by correct calculation seen within CI formula |
| \(t_{11} = 3.106\) | B1 (AO 1.1b) | AWRT \(3.106\) |
| \(86.5 \pm 3.106 \times \sqrt{\dfrac{28.45}{12}}\) | M1 (AO 1.1a) | Uses correct CI formula with correct \(\bar{x}\), \(s^2\) and \(t\) values; condone use of AWRT \(2.58\) instead of \(t\) value |
| CI is \((81.7,\ 91.3)\) | R1 (AO 2.1) | Completes reasoned argument by substituting correct values into correct formula |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| The null hypothesis is accepted as \(85\) lies within the confidence interval | E1 (AO 2.2b) | Infers null hypothesis accepted as \(85\) lies within CI; must see \(85\) or reference to proposed population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Insufficient evidence to suggest that the mean mass of apples is different from \(85\) grams | E1 (AO 3.2a) | Concludes in context; must refer to mean mass of apples; conclusion must not be definite |
## Question 5(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\bar{x} = 86.5$, $s^2 = 28.45$ | B1 (AO 1.1b) | $s^2$ **AWRT** $28.45$ or $28.455$ (3 d.p.); or $s$ **AWRT** $5.334$; PI by correct calculation seen within CI formula |
| $t_{11} = 3.106$ | B1 (AO 1.1b) | **AWRT** $3.106$ |
| $86.5 \pm 3.106 \times \sqrt{\dfrac{28.45}{12}}$ | M1 (AO 1.1a) | Uses correct CI formula with correct $\bar{x}$, $s^2$ and $t$ values; condone use of **AWRT** $2.58$ instead of $t$ value |
| CI is $(81.7,\ 91.3)$ | R1 (AO 2.1) | Completes reasoned argument by substituting correct values into correct formula |
## Question 5(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| The null hypothesis is accepted as $85$ lies within the confidence interval | E1 (AO 2.2b) | Infers null hypothesis accepted as $85$ lies within CI; must see $85$ or reference to proposed population mean |
## Question 5(c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Insufficient evidence to suggest that the mean mass of apples is different from $85$ grams | E1 (AO 3.2a) | Concludes in context; must refer to mean mass of apples; conclusion must not be definite |5 The mass, $X$, in grams of a particular type of apple is modelled using a normal distribution.
A random sample of 12 apples is collected and the summarised results are
$$\sum x = 1038 \quad \text { and } \quad \sum x ^ { 2 } = 90100$$
5
\begin{enumerate}[label=(\alph*)]
\item A 99\% confidence interval for the population mean of the masses of the apples is constructed using the random sample.
Show that the confidence interval is $( 81.7,91.3 )$ with values correct to three significant figures.\\
5
\item Padraig claims that the population mean mass of the apples is 85 grams.
He carries out a hypothesis test at the $1 \%$ level of significance using the random sample of 12 apples.
The hypotheses are
$$\begin{aligned}
& \mathrm { H } _ { 0 } : \mu = 85 \\
& \mathrm { H } _ { 1 } : \mu \neq 85
\end{aligned}$$
State, with a reason, whether the null hypothesis is accepted or rejected.\\
5
\item Interpret, in context, the conclusion to the hypothesis test in part (b).
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Statistics 2022 Q5 [6]}}