| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2018 |
| Session | March |
| Marks | 8 |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Find minimum/maximum n for probability condition |
| Difficulty | Standard +0.3 This is a straightforward application of binomial-to-normal approximation with standard continuity correction. Part (i) requires basic recall of binomial mean and variance formulas. Part (ii) is a routine inverse normal calculation. Part (iii) requires cumulative binomial calculations but is computationally straightforward with tables/calculator. The question is slightly above average difficulty only because it requires careful application of continuity correction and comparison between approximate and exact methods, but involves no novel problem-solving or conceptual challenges. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.02d Binomial: mean np and variance np(1-p)5.04b Linear combinations: of normal distributions |
| Answer | Marks |
|---|---|
| 20 | B1 |
| Answer | Marks |
|---|---|
| 16 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(N(20, 16)\) and \(P(X < x) = 0.98\) | M1 | Correct calc" seen or implied |
| \([z = 28.7]\) | A1 | 28.7 seen or implied |
| \(n = 29\) | A1 | Allow \(n \geq 29\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(B(100, 0.2)\): \(P(\geq 29) = 0.2002\), \(P(\geq 30) = 0.01125\) | M1 | One probability from B(100, 0.2) |
| A1 | Both correct | |
| So correct answer is \(n = 30\) | A1 | Correct conclusion, cwo |
## Part (i)(a)
20 | B1 |
## Part (i)(b)
16 | B1 |
## Part (ii)
$N(20, 16)$ and $P(X < x) = 0.98$ | M1 | Correct calc" seen or implied
$[z = 28.7]$ | A1 | 28.7 seen or implied
$n = 29$ | A1 | Allow $n \geq 29$
## Part (iii)
$B(100, 0.2)$: $P(\geq 29) = 0.2002$, $P(\geq 30) = 0.01125$ | M1 | One probability from B(100, 0.2)
| A1 | Both correct
So correct answer is $n = 30$ | A1 | Correct conclusion, cwo5 A spinner has 5 edges. Each edge is numbered with a different integer from 1 to 5 . When the spinner is spun, it is equally likely to come to rest on any one of the edges. The spinner is spun 100 times. The number of times on which the spinner comes to rest on the edge numbered 5 is denoted by $X$.
\begin{enumerate}[label=(\alph*)]
\item $\mathrm { E } ( X )$,
\item $\operatorname { Var } ( X )$.
\begin{enumerate}[label=(\roman*)]
\item Write down
\item Use a normal distribution with the same mean and variance as in your answers to part (i) to estimate the smallest value of $n$ such that $\mathrm { P } ( X \geqslant n ) < 0.02$.
\item Use the binomial distribution to find exactly the smallest value of $n$ such that $\mathrm { P } ( X \geqslant n ) < 0.02$. Show the values of all relevant calculations.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2018 Q5 [8]}}