CAIE S1 2011 November — Question 1 5 marks

Exam BoardCAIE
ModuleS1 (Statistics 1)
Year2011
SessionNovember
Marks5
PaperDownload PDF ↗
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 requiring only: (1) checking conditions are met, (2) calculating mean and variance, (3) applying continuity correction, and (4) using normal tables. It's a single probability calculation with no conceptual challenges, making it easier than average but not trivial due to the multi-step process.
Spec2.04b Binomial distribution: as model B(n,p)2.04d Normal approximation to binomial
1 When a butternut squash seed is sown the probability that it will germinate is 0.86 , independently of any other seeds. A market gardener sows 250 of these seeds. Use a suitable approximation to find the probability that more than 210 germinate.
Question 1:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(\mu = 250 \times 0.86 = 215\)B1 \(250 \times 0.86\) and \(250 \times 0.86 \times 0.14\) seen
\(\sigma^2 = 250 \times 0.86 \times 0.14 = 30.1\)M1 Standardising, with or without cc, must have sq rt in denom
\(P(X > 210) = 1 - \Phi\left(\frac{210.5 - 215}{\sqrt{30.1}}\right)\)M1 Continuity correction 210.5 or 209.5 only
\(= \Phi(0.820)\)M1 Correct region \((> 0.5)\) ft their mean
\(= 0.794\)A1 [5] Correct answer 
## Question 1:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\mu = 250 \times 0.86 = 215$ | B1 | $250 \times 0.86$ and $250 \times 0.86 \times 0.14$ seen |
| $\sigma^2 = 250 \times 0.86 \times 0.14 = 30.1$ | M1 | Standardising, with or without cc, must have sq rt in denom |
| $P(X > 210) = 1 - \Phi\left(\frac{210.5 - 215}{\sqrt{30.1}}\right)$ | M1 | Continuity correction 210.5 or 209.5 only |
| $= \Phi(0.820)$ | M1 | Correct region $(> 0.5)$ ft their mean |
| $= 0.794$ | A1 [5] | Correct answer |

---
1 When a butternut squash seed is sown the probability that it will germinate is 0.86 , independently of any other seeds. A market gardener sows 250 of these seeds. Use a suitable approximation to find the probability that more than 210 germinate.

\hfill \mbox{\textit{CAIE S1 2011 Q1 [5]}}
This paper (7 questions)
View full paper
Q1 5 Q2 6 Q3 6 Q4 6 Q5 9 Q6 9 Q7 9