CAIE FP2 2010 June — Question 6 5 marks

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2010
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicNormal Distribution
TypeLinear transformation of normal
DifficultyModerate -0.3 This question requires understanding that the median of log₁₀X equals its mean (1.5), then exponentiating to get X = 10^1.5. The probability calculation is a straightforward standardization: P(X ≥ 50) = P(log₁₀X ≥ log₁₀50) followed by a z-score lookup. While it tests conceptual understanding of log-normal distributions, the mechanics are routine once the transformation is recognized.
Spec2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation
6 The lifetime, \(X\) days, of a particular insect is such that \(\log _ { 10 } X\) has a normal distribution with mean 1.5 and standard deviation 0.2. Find the median lifetime. Find also \(\mathrm { P } ( X \geq 50 )\).
Question 6:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(\log_{10} M = 1.5\)M1 A1 Find relation for median \(M\)
\(M = 10^{1.5} = 31.6\)A1 Evaluate \(M\)
\(P(X \geq 50) = P(\log X \geq \log 50) = 1 - \Phi((\log 50 - 1.5)/0.2)\)M1 Relate \(P(X \geq 50)\) to Normal distribution
\([\log 50 = 1.699]\): \(= 1 - \Phi(0.995) = 0.160\)A1
Total: 5 marks
## Question 6:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\log_{10} M = 1.5$ | M1 A1 | Find relation for median $M$ |
| $M = 10^{1.5} = 31.6$ | A1 | Evaluate $M$ |
| $P(X \geq 50) = P(\log X \geq \log 50) = 1 - \Phi((\log 50 - 1.5)/0.2)$ | M1 | Relate $P(X \geq 50)$ to Normal distribution |
| $[\log 50 = 1.699]$: $= 1 - \Phi(0.995) = 0.160$ | A1 | |

**Total: 5 marks**

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6 The lifetime, $X$ days, of a particular insect is such that $\log _ { 10 } X$ has a normal distribution with mean 1.5 and standard deviation 0.2. Find the median lifetime.

Find also $\mathrm { P } ( X \geq 50 )$.

\hfill \mbox{\textit{CAIE FP2 2010 Q6 [5]}}
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