| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Multiple independent binomial calculations |
| Difficulty | Moderate -0.3 This is a straightforward S1 binomial distribution question requiring standard calculations: (a) direct use of binomial probability formula, (b) cumulative probability lookups from tables, and (c) basic comparison of sample statistics with theoretical mean/variance. All parts are routine applications of textbook methods with no problem-solving insight required, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(X \sim B(26, 0.06)\) | M1 | Correct binomial setup |
| \(P(X=2) = \binom{26}{2}(0.06)^2(0.94)^{24}\) | M1 | Correct probability calculation |
| \(= 0.2249...\) awrt \(0.225\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(Y \sim B(50, 0.15)\) | Stated or implied | |
| (i) \(P(Y < 10) = P(Y \leq 9)\) | M1 | |
| \(= 0.9084\) awrt \(0.908\) | A1 | |
| (ii) \(P(Y > 5) = 1 - P(Y \leq 5)\) | M1 | |
| \(= 1 - 0.4465 = 0.5535\) awrt \(0.554\) | A1 | |
| (iii) \(P(6 < Y < 12) = P(Y \leq 11) - P(Y \leq 6)\) | M1 | |
| \(= 0.9468 - 0.5688 = 0.3780\) awrt \(0.378\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Expected value for \(B(50, 0.15)\): \(np = 50 \times 0.15 = 7.5\) | B1 | Correct mean for chain |
| \(\bar{x} = 4.33 < 7.5\) so Farokh's store has fewer incomplete orders on average | B1 | Comparison of means with context |
| Expected variance: \(np(1-p) = 50 \times 0.15 \times 0.85 = 6.375\) | B1 | Correct variance for chain |
| \(s^2 = 3.94 < 6.375\) so less variability in Farokh's store | B1 | Comparison of variances with context |
| Overall: Farokh's store performs better than the chain as a whole | B1 | Conclusion in context |
# Question 6:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $X \sim B(26, 0.06)$ | M1 | Correct binomial setup |
| $P(X=2) = \binom{26}{2}(0.06)^2(0.94)^{24}$ | M1 | Correct probability calculation |
| $= 0.2249...$ awrt $0.225$ | A1 | |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $Y \sim B(50, 0.15)$ | | Stated or implied |
| **(i)** $P(Y < 10) = P(Y \leq 9)$ | M1 | |
| $= 0.9084$ awrt $0.908$ | A1 | |
| **(ii)** $P(Y > 5) = 1 - P(Y \leq 5)$ | M1 | |
| $= 1 - 0.4465 = 0.5535$ awrt $0.554$ | A1 | |
| **(iii)** $P(6 < Y < 12) = P(Y \leq 11) - P(Y \leq 6)$ | M1 | |
| $= 0.9468 - 0.5688 = 0.3780$ awrt $0.378$ | A1 | |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| Expected value for $B(50, 0.15)$: $np = 50 \times 0.15 = 7.5$ | B1 | Correct mean for chain |
| $\bar{x} = 4.33 < 7.5$ so Farokh's store has fewer incomplete orders on average | B1 | Comparison of means with context |
| Expected variance: $np(1-p) = 50 \times 0.15 \times 0.85 = 6.375$ | B1 | Correct variance for chain |
| $s^2 = 3.94 < 6.375$ so less variability in Farokh's store | B1 | Comparison of variances with context |
| Overall: Farokh's store performs better than the chain as a whole | B1 | Conclusion in context |6 The probability that an online order from a supermarket chain has at least one item missing when delivered is 0.06 .
Online orders are 'incomplete' if they contain substitute items and/or have at least one item missing when delivered. The probability that an order is incomplete is 0.15 .
\begin{enumerate}[label=(\alph*)]
\item Calculate the probability that exactly 2 out of a random sample of 26 online orders have at least one item missing when delivered.
\item Determine the probability that the number of incomplete orders in a random sample of 50 online orders is:
\begin{enumerate}[label=(\roman*)]
\item fewer than 10 ;
\item more than 5;
\item more than 6 but fewer than 12 .
\end{enumerate}\item Farokh, the manager of one of the supermarket's stores, examines 50 randomly selected online orders from each of a random sample of 100 of the store's customers. He records, for each of the 50 orders, the number, $x$, that were incomplete.
His summarised results, correct to three significant figures, for the 100 customers selected are
$$\bar { x } = 4.33 \text { and } s ^ { 2 } = 3.94$$
Use this information to compare the performance of the store managed by Farokh with that of the supermarket chain as a whole.\\[0pt]
[5 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA S1 2014 Q6 [14]}}