| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2007 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Find parameter from normal approximation |
| Difficulty | Standard +0.3 This is a standard S2 question on normal approximation to binomial. Part (a) requires recall of the np>5 and nq>5 rule. Part (b) involves routine inverse normal calculations and solving simultaneous equations for n and p using mean=np and variance=np(1-p). While multi-step, it follows a well-practiced procedure with no novel insight required, making it slightly easier than average. |
| Spec | 2.04d Normal approximation to binomial5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(np > 5\) and \(nq > 5\); \(0.75n > 5\) is relevant; \(n ≥ 20\) | M2 | Use either \(nq > 5\) or \(npq > 5\); or "n = 20" seen: M1] [SR: If M0, use \(np\) > 5, "n = 20" only; or "n ≥ 20 only" |
| A1 | Final answer \(n ≥ 20\) or \(n = 20\) only | 3 |
| (b) (i) \(70.5 - \mu = 1.75\sigma\); \(\mu - 46.5 = 2.25\sigma\); Solve simultaneously; \(\mu = 60\); \(\sigma = ±154.5\) | M1 | Standardise once, and equate to Φ⁻¹, ± cc |
| A1 | Standardise twice, signs correct, cc correct | |
| B1 | Both 1.75 and 2.25 | |
| M1 | Correct solution method to get one variable | |
| A1 ∀ | μ, a.r.t. 60.0 or ± 154.5 | |
| A1 ∀ | σ, a.r.t. 6.00 [Wrong cc (below): A1 both] [SR: σ⁻¹: M1A0B1M1A1A0] | 6 |
| (ii) \(np = 60, npq = 36\) | M1 dep | \(np = 60\) and \(npq = 6^2\) or 6 |
| dep M1 | Solve to get \(q\) or \(p\) or \(n\) | |
| A1 ∀ | \(p = 0.4\) ∀ on wrong cc or ≥ | |
| A1 ∀ | \(n = 150\) ∀ on wrong cc or ≥ | 4 |
(a) $np > 5$ and $nq > 5$; $0.75n > 5$ is relevant; $n ≥ 20$ | M2 | Use either $nq > 5$ or $npq > 5$; or "n = 20" seen: M1] [SR: If M0, use $np$ > 5, "n = 20" only; or "n ≥ 20 only"
| A1 | Final answer $n ≥ 20$ or $n = 20$ only | 3
(b) (i) $70.5 - \mu = 1.75\sigma$; $\mu - 46.5 = 2.25\sigma$; Solve simultaneously; $\mu = 60$; $\sigma = ±154.5$ | M1 | Standardise once, and equate to Φ⁻¹, ± cc
| A1 | Standardise twice, signs correct, cc correct
| B1 | Both 1.75 and 2.25
| M1 | Correct solution method to get one variable
| A1 ∀ | μ, a.r.t. 60.0 or ± 154.5
| A1 ∀ | σ, a.r.t. 6.00 [Wrong cc (below): A1 both] [SR: σ⁻¹: M1A0B1M1A1A0] | 6
(ii) $np = 60, npq = 36$ | M1 dep | $np = 60$ and $npq = 6^2$ or 6
| dep M1 | Solve to get $q$ or $p$ or $n$
| A1 ∀ | $p = 0.4$ ∀ on wrong cc or ≥
| A1 ∀ | $n = 150$ ∀ on wrong cc or ≥ | 49
\begin{enumerate}[label=(\alph*)]
\item The random variable $G$ has the distribution $\mathrm { B } ( n , 0.75 )$. Find the set of values of $n$ for which the distribution of $G$ can be well approximated by a normal distribution.
\item The random variable $H$ has the distribution $\mathrm { B } ( n , p )$. It is given that, using a normal approximation, $\mathrm { P } ( H \geqslant 71 ) = 0.0401$ and $\mathrm { P } ( H \leqslant 46 ) = 0.0122$.
\begin{enumerate}[label=(\roman*)]
\item Find the mean and standard deviation of the approximating normal distribution.
\item Hence find the values of $n$ and $p$.
4
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR S2 2007 Q9 [13]}}