| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Probability of range of values |
| Difficulty | Moderate -0.3 This is a straightforward S1 binomial distribution question requiring standard calculations: cumulative probabilities using tables/calculator, mean/variance formulas (np and np(1-p)), and basic comparison of theoretical vs observed statistics. All techniques are routine for A-level stats 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 |
| Answer | Marks | Guidance |
|---|---|---|
| \(E \sim B(40,\ 0.30)\) | M1 | Used anywhere in (a) even only by implication from a correct value |
| \(P(E \leq 10) = 0.308\) to \(0.309\) | A1 | AWFW (0.3087) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(E \geq 15) = 1 - (0.8074 \text{ or } 0.8849)\) | M1 | Requires '\(1-\)'; accept 3 dp rounding or truncation; can be implied by 0.192 to 0.193 but not by 0.115 to 0.116 |
| \(= 0.192\) to \(0.193\) | A1 | AWFW (0.1926) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(E \leq 12) = 0.5772 - 0.4406\) or \(P(E \leq 12) = \dbinom{40}{12}0.3^{12}0.7^{28}\) | M1 | Accept 3 dp rounding or truncation; Correct expression; may be implied by a correct answer |
| \(= 0.136\) to \(0.138\) | A1 | AWFW (0.1366) |
| Answer | Marks | Guidance |
|---|---|---|
| Means \(= 3.2\) and \(2\) | B1 | CAO both values; ignore notation; if not labelled, assume order in question |
| Variances \(= 2.56\) and \(1.75\) | B1 B1 | CAO each value; ignore notation; ISW all subsequent working |
| Answer | Marks | Guidance |
|---|---|---|
| Mean \(= 2\) | B1 | CAO value; ignore notation |
| Variance \(= 2.54\) to \(2.55\) or \(2.33\) to \(2.34\) \((SD = 1.59 \text{ to } 1.6 \text{ or } 1.52 \text{ to } 1.53)\) | B1 | Any value within either range; ignore notation; ISW all subsequent working |
| Answer | Marks | Guidance |
|---|---|---|
| \(B(16, 0.20)\) or eg "One dist\(^{n}\)"; Different/larger mean; Similar/same variance or standard deviation | Bdep1 | Identification of distribution not required; Both; dep on 3.2, 2.56/1.6 & (c)(i) |
| \(B(16, 0.125)\) or eg "Other dist\(^{n}\)"; Equal/same mean; Different/smaller variance or standard deviation | Bdep1 | Identification of distribution not required; Both; dep on 2, 1.75/1.3 & (c)(i) |
| Neither likely to provide satisfactory model | Bdep1 | Dep on Bdep1 and on Bdep1 |
# Question 3:
## Part (a)(i)
$E \sim B(40,\ 0.30)$ | M1 | Used anywhere in (a) even only by implication from a correct value
$P(E \leq 10) = 0.308$ **to** $0.309$ | A1 | AWFW (0.3087)
**SC:** For calc$^n$ of individual terms: award B2 for answer within above range; award B1 for answer within range 0.3 to 0.32
## Part (a)(ii)
$P(E \geq 15) = 1 - (0.8074 \text{ or } 0.8849)$ | M1 | Requires '$1-$'; accept 3 dp rounding or truncation; can be implied by 0.192 to 0.193 but **not** by 0.115 to 0.116
$= 0.192$ **to** $0.193$ | A1 | AWFW (0.1926)
**SC:** For calc$^n$ of individual terms: award B2 for answer within above range; award B1 for answer within range 0.18 to 0.2
## Part (a)(iii)
$P(E \leq 12) = 0.5772 - 0.4406$ **or** $P(E \leq 12) = \dbinom{40}{12}0.3^{12}0.7^{28}$ | M1 | Accept 3 dp rounding or truncation; Correct expression; may be implied by a correct answer
$= 0.136$ **to** $0.138$ | A1 | AWFW (0.1366)
## Part (b)
Means $= 3.2$ **and** $2$ | B1 | CAO both **values**; ignore notation; if not labelled, assume order in question
Variances $= 2.56$ **and** $1.75$ | B1 B1 | CAO each **value**; ignore notation; ISW all subsequent working
## Part (c)(i)
Mean $= 2$ | B1 | CAO **value**; ignore notation
Variance $= 2.54$ **to** $2.55$ **or** $2.33$ **to** $2.34$ $(SD = 1.59 \text{ to } 1.6 \text{ or } 1.52 \text{ to } 1.53)$ | B1 | Any **value** within either range; ignore notation; ISW all subsequent working
## Part (c)(ii)
$B(16, 0.20)$ **or** eg "One dist$^{n}$"; **Different/larger** mean; **Similar/same** variance or standard deviation | Bdep1 | Identification of distribution **not** required; Both; dep on 3.2, 2.56/1.6 & (c)(i)
$B(16, 0.125)$ **or** eg "Other dist$^{n}$"; **Equal/same** mean; **Different/smaller** variance or standard deviation | Bdep1 | Identification of distribution **not** required; Both; dep on 2, 1.75/1.3 & (c)(i)
**Neither** likely to provide satisfactory model | Bdep1 | Dep on Bdep1 and on Bdep13 Stopoff owns a chain of hotels. Guests are presented with the bills for their stays when they check out.
\begin{enumerate}[label=(\alph*)]
\item Assume that the number of bills that contain errors may be modelled by a binomial distribution with parameters $n$ and $p$, where $p = 0.30$.
Determine the probability that, in a random sample of 40 bills:
\begin{enumerate}[label=(\roman*)]
\item at most 10 bills contain errors;
\item at least 15 bills contain errors;
\item exactly 12 bills contain errors.
\end{enumerate}\item Calculate the mean and the variance for each of the distributions $\mathrm { B } ( 16,0.20 )$ and $B ( 16,0.125 )$.
\item Stan, who is a travelling salesperson, always uses Stopoff hotels. He holds one of its diamond customer cards and so should qualify for special customer care. However, he regularly finds errors in his bills when he checks out.
Each month, during a 12-month period, Stan stayed in Stopoff hotels on exactly 16 occasions. He recorded, each month, the number of occasions on which his bill contained errors. His recorded values were as follows.
$$\begin{array} { l l l l l l l l l l l l }
2 & 1 & 4 & 3 & 1 & 3 & 0 & 3 & 1 & 0 & 5 & 1
\end{array}$$
\begin{enumerate}[label=(\roman*)]
\item Calculate the mean and the variance of these 12 values.
\item Hence state with reasons which, if either, of the distributions $\mathrm { B } ( 16,0.20 )$ and $B ( 16,0.125 )$ is likely to provide a satisfactory model for these 12 values.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S1 2013 Q3 [14]}}