| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2007 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Probability of range of values |
| Difficulty | Standard +0.3 This is a straightforward binomial distribution question requiring standard calculations: P(X ≤ 3) and P(10 < X < 20) using tables, finding mean/SD using np and √(np(1-p)), calculating sample statistics, and a basic comparison comment. All techniques are routine S1 content with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial |
| 2 | 2 | 4 | 1 | 2 | 3 | 3 | 2 | 2 | 0 | 3 | 2 | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use of binomial in (a) or (b)(i) | M1 | PI |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(T_{10} \leq 3) = 0.38\) to \(0.383\) | B1 | AWFW; \((0.3823)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(10 < T_{40} < 20) = 0.8702\) or \(0.9256\) | M1 | Allow 3 dp accuracy |
| minus \(0.0352\) or \(0.0156\) | M1 | Allow 3 dp accuracy |
| \(= 0.83\) to \(0.84\) | A1 | AWFW; \((0.835)\) |
| OR \(B(40, 0.40)\) expressions stated for at least 3 terms within \(10 \leq T_{40} \leq 20\) | (M1) | Or implied by a correct answer |
| Answer \(= 0.83\) to \(0.84\) | (A2) | AWFW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(n = 5\), \(p = 0.4\) | — | — |
| Mean, \(\mu = np = 2\) | B1 | CAO |
| Variance, \(\sigma^2 = np(1-p) = 1.2\) | M1 | Use of \(np(1-p)\) even if SD |
| Standard deviation \(= \sqrt{1.2} = 1.09\) to \(1.1\) | A1 | CAO; AWFW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Mean \((\bar{x}) = 2\) | B1 | CAO; \(\sum x = 26\) |
| Standard Deviation \((s_n, s_{n-1}) = 1.1\) to \(1.16\) | B2 | AWFW; \(\sum x^2 = 68\); \((1.1094\) or \(1.1547)\) |
| If neither correct but use of mean \((\bar{x}) = \frac{\sum x}{13}\) | (M1) | — |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Means are same and SDs are similar/same or Means are same but SDs are different | B1 | Must have scored full marks in (b)(i) and (b)(ii) |
| so Trina's claims appear valid/invalid | ↑Dep↑ B1 | — |
# Question 6:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| Use of binomial in (a) or (b)(i) | M1 | PI |
## Part (a)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(T_{10} \leq 3) = 0.38$ to $0.383$ | B1 | AWFW; $(0.3823)$ |
## Part (a)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(10 < T_{40} < 20) = 0.8702$ or $0.9256$ | M1 | Allow 3 dp accuracy |
| minus $0.0352$ or $0.0156$ | M1 | Allow 3 dp accuracy |
| $= 0.83$ to $0.84$ | A1 | AWFW; $(0.835)$ |
| **OR** $B(40, 0.40)$ expressions stated for at least 3 terms within $10 \leq T_{40} \leq 20$ | (M1) | Or implied by a correct answer |
| Answer $= 0.83$ to $0.84$ | (A2) | AWFW |
## Part (b)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $n = 5$, $p = 0.4$ | — | — |
| Mean, $\mu = np = 2$ | B1 | CAO |
| Variance, $\sigma^2 = np(1-p) = 1.2$ | M1 | Use of $np(1-p)$ even if SD |
| Standard deviation $= \sqrt{1.2} = 1.09$ to $1.1$ | A1 | CAO; AWFW |
## Part (b)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Mean $(\bar{x}) = 2$ | B1 | CAO; $\sum x = 26$ |
| Standard Deviation $(s_n, s_{n-1}) = 1.1$ to $1.16$ | B2 | AWFW; $\sum x^2 = 68$; $(1.1094$ or $1.1547)$ |
| If neither correct but use of mean $(\bar{x}) = \frac{\sum x}{13}$ | (M1) | — |
## Part (b)(iii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Means are same and SDs are similar/same **or** Means are same but SDs are different | B1 | Must have scored full marks in (b)(i) and (b)(ii) |
| so Trina's claims appear valid/invalid | ↑Dep↑ B1 | — |
---6 Each weekday, Monday to Friday, Trina catches a train from her local station. She claims that the probability that the train arrives on time at the station is 0.4 , and that the train's arrival time is independent from day to day.
\begin{enumerate}[label=(\alph*)]
\item Assuming her claims to be true, determine the probability that the train arrives on time at the station:
\begin{enumerate}[label=(\roman*)]
\item on at most 3 days during a 2 -week period ( 10 days);
\item on more than 10 days but fewer than 20 days during an 8-week period.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Assuming Trina's claims to be true, determine the mean and standard deviation for the number of times during a week (5 days) that the train arrives on time at the station.
\item Each week, for a period of 13 weeks, Trina's travelling colleague, Suzie, records the number of times that the train arrives on time at the station. Suzie's results are
\begin{center}
\begin{tabular}{ l l l l l l l l l l l l l }
2 & 2 & 4 & 1 & 2 & 3 & 3 & 2 & 2 & 0 & 3 & 2 & 0 \\
\end{tabular}
\end{center}
Calculate the mean and standard deviation of these values.
\item Hence comment on the likely validity of Trina's claims.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S1 2007 Q6 [13]}}