Standard +0.8 This S2 question requires working backwards from a given probability to find the sample size n, involving setting up X ~ B(n, 0.2), applying normal approximation with continuity correction, using inverse normal tables to find the z-score for P(Z > z) = 0.0401, then solving a quadratic equation in √n. The multi-step reverse-engineering and algebraic manipulation make this harder than standard forward normal approximation problems.
5. In a large school, \(20 \%\) of students own a touch screen laptop. A random sample of \(n\) students is chosen from the school. Using a normal approximation, the probability that more than 55 of these \(n\) students own a touch screen laptop is 0.0401 correct to 3 significant figures.
Find the value of \(n\).
(8)
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Using \(z =\) awrt \(\pm 1.75\); standardising using 55.5, 54.5 or 55 and equating to a \(z\) value; follow through their mean and variance; A1: correct equation; condone inequality sign
\(0.2n + 0.7\sqrt{n} - 55.5 = 0\)
M1d
Dependent on previous M; using quadratic formula, completing the square, factorising, or any correct method to solve their 3-term equation; may be implied by \(\sqrt{n} = 15\) or 342.25
\(\sqrt{n} = 15\)
A1
Allow 15 or \(-18.5\); condone \(n = 15\) or \(n = -18.5\)
\(n = 225\)
A1
cao 225
Alternative method:
Answer
Marks
Guidance
Answer/Working
Mark
Guidance
\((0.2n - 55.5)^2 = (-0.7\sqrt{n})^2\)
M1
Solving 3-term quadratic in \(n\)
\(0.04n^2 - 22.69n + 3080.25 = 0\)
\(n = 225\) or \(1369/4\); \(n = 225\)
A1
Either 225 or \(1369/4\) or 342.25; must select 225
## Question 5:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $N(0.2n,\; 0.16n)$ | B1 | Mean $= 0.2n$ and Var $= 0.16n$ oe; may be awarded if they appear in the standardisation as $0.2n$ and either $0.16n$ or $\sqrt{0.16n}$ |
| $P\!\left(Z > \frac{55.5 - 0.2n}{\sqrt{0.16n}}\right) = 0.0401$ | M1 | Using a continuity correction either 55.5 or 54.5 |
| $\frac{55.5 - 0.2n}{\sqrt{0.16n}} = 1.75$ | B1M1A1 | Using $z =$ awrt $\pm 1.75$; standardising using 55.5, 54.5 or 55 and equating to a $z$ value; follow through their mean and variance; A1: correct equation; condone inequality sign |
| $0.2n + 0.7\sqrt{n} - 55.5 = 0$ | M1d | Dependent on previous M; using quadratic formula, completing the square, factorising, or any correct method to solve their 3-term equation; may be implied by $\sqrt{n} = 15$ or 342.25 |
| $\sqrt{n} = 15$ | A1 | Allow 15 or $-18.5$; condone $n = 15$ or $n = -18.5$ |
| $n = 225$ | A1 | cao 225 |
**Alternative method:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(0.2n - 55.5)^2 = (-0.7\sqrt{n})^2$ | M1 | Solving 3-term quadratic in $n$ |
| $0.04n^2 - 22.69n + 3080.25 = 0$ | | |
| $n = 225$ or $1369/4$; $n = 225$ | A1 | Either 225 or $1369/4$ or 342.25; must select 225 |
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5. In a large school, $20 \%$ of students own a touch screen laptop. A random sample of $n$ students is chosen from the school. Using a normal approximation, the probability that more than 55 of these $n$ students own a touch screen laptop is 0.0401 correct to 3 significant figures.
Find the value of $n$.\\
(8)\\
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\hfill \mbox{\textit{Edexcel S2 2016 Q5 [8]}}