9231 P42 - Nov 2020 - Q3 - 7 marks
6835
A random sample of 200 observations of the continuous random variable \(X\) was taken and the values are summarised in the following table.
Interval | \(0 \leqslant x\lt 0.5\) | \(0.5 \leqslant x\lt 1\) | \(1 \leqslant x\lt 1.5\) | \(1.5 \leqslant x\lt 2\) | \(2 \leqslant x\lt 2.5\) | \(2.5 \leqslant x\lt 3\) |
|---|---|---|---|---|---|---|
Observed frequency | 5 | 23 | 40 | 41 | 46 | 45 |
It is required to test the goodness of fit of the distribution with probability density function f given by
\(f(x)=\left\{\begin{array}{ll} \frac{1}{9} x(4-x) & 0 \leqslant x \leqslant 3, \\ 0 & \text { otherwise } . \end{array}\right.\)
Most of the relevant expected frequencies, correct to 2 decimal places, are given in the following table.
Interval | \(0 \leqslant x\lt 0.5\) | \(0.5 \leqslant x\lt 1\) | \(1 \leqslant x\lt 1.5\) | \(1.5 \leqslant x\lt 2\) | \(2 \leqslant x\lt 2.5\) | \(2.5 \leqslant x\lt 3\) |
|---|---|---|---|---|---|---|
Expected frequency | \(p\) | \(q\) | 37.96 | 43.52 | 43.52 | 37.96 |
(a) Show that \(p=10.19\) and find the value of \(q\).
(b) Carry out a goodness of fit test, at the \(5 \%\) significance level, to test whether f is a satisfactory model for the data.
Solutions and mark schemes for 9231 are temporarily available to admins only.