CAIE S2 2012 November — Question 1 3 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2012
SessionNovember
Marks3
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicContinuous Probability Distributions and Random Variables
TypeFind median or percentiles
DifficultyModerate -0.8 This is a straightforward median calculation from a given PDF graph. Students need only to recognize that the median divides the area under the curve in half, then set up and solve a simple integral equation (or use geometric area if the shape is simple). This requires basic understanding of continuous distributions but minimal problem-solving beyond applying the definition.
Spec5.03f Relate pdf-cdf: medians and percentiles
1 \includegraphics[max width=\textwidth, alt={}, center]{879cb813-2380-47a7-bd96-cad0a74d0b4d-2_369_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 \(\int_0^m (\frac{1}{2}x)dx = \frac{1}{2}\)
\(m = \sqrt{2}\) 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 | $\int_0^m (\frac{1}{2}x)dx = \frac{1}{2}$
$m = \sqrt{2}$ or $1.41$ (3 sfs) | A1 [3] | (Note: $\pm\sqrt{2}$ as final answer scores A0)
1\\
\includegraphics[max width=\textwidth, alt={}, center]{879cb813-2380-47a7-bd96-cad0a74d0b4d-2_369_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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