OCR S2 — Question 2 7 marks

Exam BoardOCR
ModuleS2 (Statistics 2)
Marks7
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Mark schemeDownload PDF ↗
TopicApproximating Binomial to Normal Distribution
TypeSingle probability inequality
DifficultyModerate -0.8 This is a straightforward application of the normal approximation to the binomial distribution with continuity correction. Students need to calculate mean and variance, apply the continuity correction (P(W > 13) = P(W ≥ 14) becomes P(Z > 13.5)), then use standard normal tables. It's a routine S2 question requiring only standard procedure with no problem-solving or conceptual challenges beyond remembering the continuity correction.
Spec2.04d Normal approximation to binomial5.04a Linear combinations: E(aX+bY), Var(aX+bY)
2 The random variable \(W\) has the distribution \(B \left( 40 , \frac { 2 } { 7 } \right)\). Use an appropriate approximation to find \(\mathrm { P } ( W > 13 )\).
Answer: \(\text{N}(80/7, 400/49)\)
\(13.5 - \frac{80}{7} = \frac{-5.5}{\sqrt{20}}\)
\(= 0.725\)
AnswerMarks Guidance
\(1 - \Phi(0.725) = 0.2343\)Marks: B1 B1 M1 A1 A1 M1 A1 Guidance: Standardise with \(np\) & \(npq\) or \(\sqrt{npq}\) or \(nq\), no \(\sqrt{n}\). \(\sqrt{npq}\) correct. 13.5 correct. Normal tables used, answer < 0.5. Answer, a.r.t. 0.234. [SR: Binomial, complete expression M1, 0.231 A1; Po(80/7) B1, complete expression M1, 0.260 A1; Normal approx to Poisson, B1B0 M1A0A1 M1A0] 
**Answer:** $\text{N}(80/7, 400/49)$
$13.5 - \frac{80}{7} = \frac{-5.5}{\sqrt{20}}$
$= 0.725$
$1 - \Phi(0.725) = 0.2343$ | **Marks:** B1 B1 M1 A1 A1 M1 A1 | **Guidance:** Standardise with $np$ & $npq$ or $\sqrt{npq}$ or $nq$, no $\sqrt{n}$. $\sqrt{npq}$ correct. 13.5 correct. Normal tables used, answer < 0.5. Answer, a.r.t. 0.234. [SR: Binomial, complete expression M1, 0.231 A1; Po(80/7) B1, complete expression M1, 0.260 A1; Normal approx to Poisson, B1B0 M1A0A1 M1A0]
2 The random variable $W$ has the distribution $B \left( 40 , \frac { 2 } { 7 } \right)$. Use an appropriate approximation to find $\mathrm { P } ( W > 13 )$.

\hfill \mbox{\textit{OCR S2  Q2 [7]}}
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