| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Find minimum/maximum n for probability condition |
| Difficulty | Standard +0.8 This question requires setting up a binomial-to-normal approximation with continuity correction, then working backwards from a probability constraint to find n. It involves multiple steps: recognizing X~B(n,0.2), applying normal approximation, using continuity correction for P(X≥30), standardizing to find the z-value corresponding to 0.0228, then solving algebraically for n. The reverse-engineering aspect and need to handle the inequality correctly makes this more challenging than standard forward probability calculations. |
| Spec | 2.04d Normal approximation to binomial5.04b Linear combinations: of normal distributions |
| VIIIV SIHI NI J14M 10N OC | VIIN SIHI NI III HM ION OO | VERV SIHI NI JIIIM ION OO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| \(\bar{Y} \sim N\left(\dfrac{n}{5}, \dfrac{4n}{25}\right)\) | B1 | Writing or using \(N\left(\dfrac{n}{5}, \dfrac{4n}{25}\right)\) |
| \(P(Y \geq 30) = P\left(Z > \dfrac{29.5 - \frac{n}{5}}{\frac{2}{5}\sqrt{n}}\right)\) | M1 | Writing or using \(30 +/- 0.5\) |
| M1 | Standardising using 29, 29.5, 30 or 30.5 and their mean and their sd | |
| A1 | Fully correct standardisation (allow \(+/-\)) | |
| \(\dfrac{29.5 - \frac{n}{5}}{\frac{2}{5}\sqrt{n}} = 2\) | B1 | For \(z = +/- 2\) or awrt 2.00 must be compatible with their standardisation |
| \(n + 4\sqrt{n} - 147.5 = 0\) or \(0.04n^2 - 12.44n + 870.25 = 0\) | dM1 | (Dependent on 2nd M1) getting quadratic equation and solving leading to a value of \(\sqrt{n}\) or \(n\) |
| \(\sqrt{n} = 10.3\ldots \quad n = 106.26\ldots\) or \(n = 204.73\ldots\) | A1 | Awrt 10.3 or awrt (106 or 107 or 204 or 205) |
| \(n = 106\) | A1 cao | For 106 only (must reject other solutions if stated) |
# Question 7:
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $\bar{Y} \sim N\left(\dfrac{n}{5}, \dfrac{4n}{25}\right)$ | B1 | Writing or using $N\left(\dfrac{n}{5}, \dfrac{4n}{25}\right)$ |
| $P(Y \geq 30) = P\left(Z > \dfrac{29.5 - \frac{n}{5}}{\frac{2}{5}\sqrt{n}}\right)$ | M1 | Writing or using $30 +/- 0.5$ |
| | M1 | Standardising using 29, 29.5, 30 or 30.5 and their mean and their sd |
| | A1 | Fully correct standardisation (allow $+/-$) |
| $\dfrac{29.5 - \frac{n}{5}}{\frac{2}{5}\sqrt{n}} = 2$ | B1 | For $z = +/- 2$ or awrt 2.00 must be compatible with their standardisation |
| $n + 4\sqrt{n} - 147.5 = 0$ or $0.04n^2 - 12.44n + 870.25 = 0$ | dM1 | (Dependent on 2nd M1) getting quadratic equation **and** solving leading to a value of $\sqrt{n}$ or $n$ |
| $\sqrt{n} = 10.3\ldots \quad n = 106.26\ldots$ or $n = 204.73\ldots$ | A1 | Awrt 10.3 **or** awrt (106 **or** 107 **or** 204 **or** 205) |
| $n = 106$ | A1 cao | For 106 only (must reject other solutions if stated) |
**Note:** $\dfrac{29.5 - \frac{n}{5}}{\frac{2}{5}\sqrt{n}} = -2$ leading to an answer of 106 may score B1M1M1A1B0M1A1A1
**(8 marks)**\begin{enumerate}
\item A multiple choice examination paper has $n$ questions where $n > 30$
\end{enumerate}
Each question has 5 answers of which only 1 is correct. A pass on the paper is obtained by answering 30 or more questions correctly.
The probability of obtaining a pass by randomly guessing the answer to each question should not exceed 0.0228
Use a normal approximation to work out the greatest number of questions that could be used.\\
7.
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VIIIV SIHI NI J14M 10N OC & VIIN SIHI NI III HM ION OO & VERV SIHI NI JIIIM ION OO \\
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\hfill \mbox{\textit{Edexcel S2 2018 Q7 [8]}}