Standard +0.3 This is a standard binomial approximation question requiring students to verify conditions for normal approximation (np and nq both >5), apply continuity correction, and use normal tables. While it involves multiple steps (checking validity, standardizing with continuity correction, looking up z-value), it's a routine S2 procedure with no conceptual difficulty beyond textbook exercises.
The random variable \(F\) has the distribution B\((40, 0.65)\). Use a suitable approximation to find P\((F \leq 30)\), justifying your approximation. [7]
Standardise, their \(np\), no \(\sqrt{}\) issues, allow cc or \(\sqrt{}\) errors
A1
cc and \(\sqrt{(\text{their } np)}\) correct
\(= \mathbf{0.9322}\)
B1
Final answer, a.r.t. 0.932
"\(np > 5\)" or "\(n\) large" stated
B1
One condition asserted
"\(14 > 5\)" or "\(p\) close to \(\frac{1}{2}\)" stated
B1
Complementary condition, if "\(nq\)" must see 14 somewhere. Not \(npq\) (7 marks)
Extra conditions, e.g. "\(n > 30\)": max B1B0
SC: Exact \((0.935564)\): maximum B1B1
$N(26, 9.1)$ | M1 | Normal, mean their attempt at $40 \times 0.65$
$\Phi\left(\frac{30.5 - 26}{\sqrt{9.1}}\right) = \Phi(1.492)$ | A1 | Mean 26 and variance or SD 9.1
| M1 | Standardise, their $np$, no $\sqrt{}$ issues, allow cc or $\sqrt{}$ errors
| A1 | cc and $\sqrt{(\text{their } np)}$ correct
$= \mathbf{0.9322}$ | B1 | Final answer, a.r.t. 0.932
"$np > 5$" or "$n$ large" stated | B1 | One condition asserted
"$14 > 5$" or "$p$ close to $\frac{1}{2}$" stated | B1 | Complementary condition, if "$nq$" must see 14 somewhere. Not $npq$ (7 marks)
| | Extra conditions, e.g. "$n > 30$": max B1B0
| | SC: Exact $(0.935564)$: maximum B1B1
The random variable $F$ has the distribution B$(40, 0.65)$. Use a suitable approximation to find P$(F \leq 30)$, justifying your approximation. [7]
\hfill \mbox{\textit{OCR S2 2016 Q3 [7]}}