Edexcel S2 2015 January — Question 7 8 marks

Exam BoardEdexcel
ModuleS2 (Statistics 2)
Year2015
SessionJanuary
Marks8
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TopicApproximating Binomial to Normal Distribution
TypeFind minimum/maximum n for probability condition
DifficultyStandard +0.8 This question requires setting up a binomial-to-normal approximation with continuity correction, then working backwards from a given probability to find n using inverse normal tables. It combines multiple statistical concepts (approximation conditions, continuity correction, inequality manipulation) and requires algebraic manipulation of the standardized normal variable to isolate n, going beyond routine application of the approximation.
Spec2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial2.04f Find normal probabilities: Z transformation
7. A multiple choice examination paper has \(n\) questions where \(n > 30\) 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.
Question 7:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(Y \sim N\left(\frac{n}{5}, \frac{4n}{25}\right)\)B1 Writing or using \(N\left(\frac{n}{5}, \frac{4n}{25}\right)\)
\(P(Y \geq 30) = P\left(Z > \frac{29.5 - \frac{n}{5}}{\frac{2}{5}\sqrt{n}}\right)\)M1 Writing or using \(30 +/- 0.5\)
M1A1M1: standardising using 29, 29.5, 30 or 30.5 with their mean and sd; A1: fully correct standardisation (allow \(+/-\))
\(\frac{29.5 - \frac{n}{5}}{\frac{2}{5}\sqrt{n}} = 2\)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 \(n\)
\(\sqrt{n} = 10.3\ldots\)     \(n = 106.26\ldots\) 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)
Total: 8 marks
Note: \(\frac{29.5 - \frac{n}{5}}{\frac{2}{5}\sqrt{n}} = -2\) leading to an answer of 106 may score B1M1M1A1B0M1A1A1
## Question 7:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $Y \sim N\left(\frac{n}{5}, \frac{4n}{25}\right)$ | B1 | Writing or using $N\left(\frac{n}{5}, \frac{4n}{25}\right)$ |
| $P(Y \geq 30) = P\left(Z > \frac{29.5 - \frac{n}{5}}{\frac{2}{5}\sqrt{n}}\right)$ | M1 | Writing or using $30 +/- 0.5$ |
| | M1A1 | M1: standardising using 29, 29.5, 30 or 30.5 with their mean and sd; A1: fully correct standardisation (allow $+/-$) |
| $\frac{29.5 - \frac{n}{5}}{\frac{2}{5}\sqrt{n}} = 2$ | 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 $n$ |
| $\sqrt{n} = 10.3\ldots$     $n = 106.26\ldots$ 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) |

**Total: 8 marks**

**Note:** $\frac{29.5 - \frac{n}{5}}{\frac{2}{5}\sqrt{n}} = -2$ leading to an answer of 106 may score B1M1M1A1B0M1A1A1
7. A multiple choice examination paper has $n$ questions where $n > 30$

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.

\hfill \mbox{\textit{Edexcel S2 2015 Q7 [8]}}
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Q1 16 Q2 11 Q3 11 Q4 7 Q5 9 Q6 13 Q7 8