9231 P21 - Jun 2019 - Q9 - 10 marks
6124
A random sample of 50 observations of the continuous random variable \(X\) was taken and the values are summarised in the following table.
| Interval | \(0 \leqslant x < 0.8\) | \(0.8 \leqslant x < 1.6\) | \(1.6 \leqslant x < 2.4\) | \(2.4 \leqslant x < 3.2\) | \(3.2 \leqslant x < 4\) |
|---|---|---|---|---|---|
| Observed frequency | 18 | 16 | 8 | 6 | 2 |
It is required to test the goodness of fit of the distribution with probability density function \(f\) given by
\[ f(x)= \begin{cases} \dfrac{3}{16}(4-x)^{\frac12}, & 0 \leqslant x < 4,\\[4pt] 0, & \text{otherwise}. \end{cases} \]
The relevant expected frequencies, correct to 2 decimal places, are given in the following table.
| Interval | \(0 \leqslant x < 0.8\) | \(0.8 \leqslant x < 1.6\) | \(1.6 \leqslant x < 2.4\) | \(2.4 \leqslant x < 3.2\) | \(3.2 \leqslant x < 4\) |
|---|---|---|---|---|---|
| Expected frequency | 14.22 | 12.54 | 10.59 | 8.18 | 4.47 |
(i) Show how the expected frequency for \(1.6 \leqslant x < 2.4\) is obtained.
(ii) Carry out a goodness of fit test at the \(5\%\) significance level.
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