CAIE S2 2012 November — Question 1 3 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2012
SessionNovember
Marks3
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Mark schemeDownload PDF ↗
TopicContinuous Uniform Random Variables
TypeInterquartile range and percentiles
DifficultyEasy -1.2 This is a straightforward application of the definition of median for a continuous uniform distribution. Students need only recognize the rectangular PDF, identify the uniform distribution on [2,6], and recall that the median of a symmetric distribution is at its center, giving 4. Requires minimal calculation and is simpler than typical A-level questions.
Spec5.03f Relate pdf-cdf: medians and percentiles
1 \includegraphics[max width=\textwidth, alt={}, center]{0cd5fc36-486d-4c24-b809-907b3e87cfd7-2_371_531_255_806} The diagram shows the graph of the probability density function, f , of a random variable \(X\). Find the median of \(X\).
AnswerMarks Guidance
\((\frac{m}{2})^2\)M1 Attempt at linear equ with \(c = 0\)
\((\frac{m}{2})^2 = \frac{1}{4}\)M1 \(\int_0^m (\frac{1}{x})dx = \frac{1}{2}\)
\(m = \sqrt{2}\) or \(1.41\) (3 sfs)A1 [3] (Note: \(\pm\sqrt{2}\) as final answer scores A0) 
$(\frac{m}{2})^2$ | M1 | Attempt at linear equ with $c = 0$
$(\frac{m}{2})^2 = \frac{1}{4}$ | M1 | $\int_0^m (\frac{1}{x})dx = \frac{1}{2}$
$m = \sqrt{2}$ or $1.41$ (3 sfs) | A1 [3] | (Note: $\pm\sqrt{2}$ as final answer scores A0)
1\\
\includegraphics[max width=\textwidth, alt={}, center]{0cd5fc36-486d-4c24-b809-907b3e87cfd7-2_371_531_255_806}

The diagram shows the graph of the probability density function, f , of a random variable $X$. Find the median of $X$.

\hfill \mbox{\textit{CAIE S2 2012 Q1 [3]}}
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