AQA S2 2011 June — Question 2

Exam BoardAQA
ModuleS2 (Statistics 2)
Year2011
SessionJune
TopicChi-squared distribution
2
  1. The continuous random variable \(X\) has a rectangular distribution defined by the probability density function $$f ( x ) = \begin{cases} 0.01 \pi & u \leqslant x \leqslant 11 u
    0 & \text { otherwise } \end{cases}$$ where \(u\) is a constant.
    1. Show that \(u = \frac { 10 } { \pi }\).
    2. Using the formulae for the mean and the variance of a rectangular distribution, find, in terms of \(\pi\), values for \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
    3. Calculate exact values for the mean and the variance of the circumferences of circles having diameters of length \(\left( X + \frac { 10 } { \pi } \right)\).
  2. A machine produces circular discs which have an area of \(Y \mathrm {~cm} ^ { 2 }\). The distribution of \(Y\) has mean \(\mu\) and variance 25 . A random sample of 100 such discs is selected. The mean area of the discs in this sample is calculated to be \(40.5 \mathrm {~cm} ^ { 2 }\). Calculate a 95\% confidence interval for \(\mu\). Emily believed that the performances of 16-year-old students in their GCSEs are associated with the schools that they attend. To investigate her belief, Emily collected data on the GCSE results for 2010 from four schools in her area. The table shows Emily's collected data, denoted by \(O _ { i }\), together with the corresponding expected frequencies, \(E _ { i }\), necessary for a \(\chi ^ { 2 }\) test.
    \multirow{2}{*}{}\(\boldsymbol { \geq } \mathbf { 5 }\) GCSEs\(\mathbf { 1 } \boldsymbol { \leqslant }\) GCSEs < \(\mathbf { 5 }\)No GCSEs
    \(O _ { i }\)\(E _ { i }\)\(O _ { i }\)\(E _ { i }\)\(O _ { i }\)\(E _ { i }\)
    Jolliffe College for the Arts187193.159390.623026.23
    Volpe Science Academy175184.439786.522425.05
    Radok Music School183183.817886.233424.96
    Bailey Language School265248.61112116.632233.76
    Emily used these values to correctly conduct a \(\chi ^ { 2 }\) test at the \(1 \%\) level of significance.
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