| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2010 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Known variance confidence intervals |
| Difficulty | Moderate -0.3 This is a straightforward application of known-variance confidence intervals with clear guidance. Part (a)(i) is routine calculation, (a)(ii)-(iii) test basic understanding of CLT conditions, (b) requires simple interpretation, and (c) tests conceptual understanding of confidence level definition. The calculations are mechanical and the conceptual parts are standard S1 material, making this slightly easier than average. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.01a Permutations and combinations: evaluate probabilities5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Hence \(1010 \pm (7.0 \text{ to } 7.1)\) or \((1003, 1017)\) | B1, B1, M1, A1F, A1dep | CAO. AWFW (2.3263). Used. Must have \(\sqrt{n}\) with \(n > 1\). F on \(\bar{x}\) and \(z\) only. CAO & AWFW (accept 7). Dependent on A1F. AWRT |
| Notes: Use of \(t_{11}(0.99) = 2.718 \Rightarrow\) maximum of B1 B0 M1 A0F A0. Use of a 'corrected' 10.5 \(\Rightarrow\) maximum of B1 B1 M1 A0F A0 | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Weight of flour in a bag (may be assumed to be) is normally distributed | B1 | Or equivalent; must refer to weight |
| Answer | Marks | Guidance |
|---|---|---|
| Any number such that \(20 \leq \text{number} \leq 50\) | B1 | Must be a single integer value. Ignore any reasoning |
| Answer | Marks | Guidance |
|---|---|---|
| Statement appears dubious/incorrect/invalid | B1F, B1, B1dep | Or equivalent. F on (a)(i). Any reference to 1010 \(\Rightarrow\) B0F. Or equivalent; but not 'some'. Dependent on both B1F and B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 2/100 or 1/50 or 0.02 or 2% | B1 | CAO; not 0.02% |
## Part (a)(i)
Mean = $\frac{12120}{12} = 1010$
98% (0.98) $\Rightarrow$ $z = 2.32 \text{ to } 2.33$
CI for $\mu$ is $\bar{x} \pm z \times \frac{\sigma}{\sqrt{n}}$
Thus $1010 \pm 2.3263 \times \frac{10.5}{\sqrt{12}}$
Hence $1010 \pm (7.0 \text{ to } 7.1)$ or $(1003, 1017)$ | B1, B1, M1, A1F, A1dep | CAO. AWFW (2.3263). Used. Must have $\sqrt{n}$ with $n > 1$. F on $\bar{x}$ and $z$ only. CAO & AWFW (accept 7). Dependent on A1F. AWRT
Notes: Use of $t_{11}(0.99) = 2.718 \Rightarrow$ maximum of B1 B0 M1 A0F A0. Use of a 'corrected' 10.5 $\Rightarrow$ maximum of B1 B1 M1 A0F A0 | 5
## Part (a)(ii)
Weight of flour in a bag (may be assumed to be) is normally distributed | B1 | Or equivalent; must refer to weight | 1
## Part (a)(iii)
Any number such that $20 \leq \text{number} \leq 50$ | B1 | Must be a single integer value. Ignore any reasoning | 1
## Part (b)
1 kg or 1000 grams is outside / below CI
or
From CI, (population) mean weight is greater than 1kg or 1000 grams
3 or 3/12 or 25% of bags in sample weigh less than 1kg or 1000 grams
Statement appears dubious/incorrect/invalid | B1F, B1, B1dep | Or equivalent. F on (a)(i). Any reference to 1010 $\Rightarrow$ B0F. Or equivalent; but not 'some'. Dependent on both B1F and B1 | 3
## Part (c)
2/100 or 1/50 or 0.02 or 2% | B1 | CAO; not 0.02% | 1
---5 In a random sample of 12 bags of flour, the weight, in grams, of flour in each bag was recorded as follows.\\
$\begin{array} { l l l l l l l l l l l l } 1011 & 995 & 1018 & 1022 & 1014 & 1005 & 1017 & 1015 & 993 & 1018 & 992 & 1020 \end{array}$
\begin{enumerate}[label=(\alph*)]
\item It may be assumed that the weight of flour in a bag is normally distributed with a standard deviation of 10.5 grams.
\begin{enumerate}[label=(\roman*)]
\item Construct a $98 \%$ confidence interval for the mean weight, $\mu$ grams, of flour in a bag, giving the limits to four significant figures.
\item State why, in constructing your confidence interval, use of the Central Limit Theorem was not necessary.
\item If the distribution of the weight of flour in a bag was unknown, indicate a minimum number of weights that you would consider necessary for a confidence interval for $\mu$ to be valid.
\end{enumerate}\item The statement ' 1 kg ' is printed on each bag.
Comment on this statement using both the confidence interval that you constructed in part (a)(i) and the weights of the given sample of 12 bags.
\item Given that $\mu = 1000$, state the probability that a $98 \%$ confidence interval for $\mu$ will not contain 1000.\\
(l mark)
\end{enumerate}
\hfill \mbox{\textit{AQA S1 2010 Q5 [11]}}