9231 P41 - Jun 2023 - Q3 - 9 marks
6675
A random sample of 50 values of the continuous random variable \(X\) was taken. These values are summarised in the following table.
Interval | \(1 \leqslant x\lt 1.5\) | \(1.5 \leqslant x\lt 2\) | \(2 \leqslant x\lt 2.5\) | \(2.5 \leqslant x\lt 3\) | \(3 \leqslant x\lt 3.5\) | \(3.5 \leqslant x \leqslant 4\) |
|---|---|---|---|---|---|---|
Observed frequency | 3 | 3 | 8 | 11 | 13 | 12 |
It is required to test the goodness of fit of the distribution with probability density function f given by
\(f(x)=\left\{\begin{array}{ll} \frac{1}{24}\left(\frac{4}{x^{2}}+x^{2}\right) & 1 \leqslant x \leqslant 4, \\ 0 & \text { otherwise } . \end{array}\right.\)
The expected frequencies, correct to 4 decimal places, are given in the following table.
Interval | \(1 \leqslant x\lt 1.5\) | \(1.5 \leqslant x\lt 2\) | \(2 \leqslant x\lt 2.5\) | \(2.5 \leqslant x\lt 3\) | \(3 \leqslant x\lt 3.5\) | \(3.5 \leqslant x \leqslant 4\) |
|---|---|---|---|---|---|---|
Expected frequency | 4.4271 | \(a\) | 6.1285 | 8.4549 | \(b\) | 14.9678 |
(a) Show that \(a=4.6007\) and find the value of \(b\).
(b) Carry out a goodness of fit test, at the \(10 \%\) significance level, to test whether f is a satisfactory model for the data.
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