A continuous random variable \(X\) is believed to have the probability density function f given by
$$f ( x ) = \begin{cases} \frac { 3 } { 10 } \left( 5 x - x ^ { 2 } - 4 \right) & 2 \leqslant x < 4
0 & \text { otherwise } \end{cases}$$
A random sample of 60 observations was taken and these values are summarised in the following grouped frequency table.
| Interval | \(2 \leqslant x < 2.4\) | \(2.4 \leqslant x < 2.8\) | \(2.8 \leqslant x < 3.2\) | \(3.2 \leqslant x < 3.6\) | \(3.6 \leqslant x < 4\) |
| Observed frequency | 19 | 17 | 16 | 8 | 0 |
The estimated mean, based on the grouped data in the table above, is 2.69 , correct to 2 decimal places. It is decided that a goodness of fit test will only be conducted if the mean predicted from the probability density function is within \(10 \%\) of the estimated mean. Show that this condition is satisfied.
The relevant expected frequencies are as follows.
| Interval | \(2 \leqslant x < 2.4\) | \(2.4 \leqslant x < 2.8\) | \(2.8 \leqslant x < 3.2\) | \(3.2 \leqslant x < 3.6\) | \(3.6 \leqslant x < 4\) |
| Expected frequency | 15.456 | 16.032 | 14.304 | 10.272 | 3.936 |
Show how the expected frequency for the interval \(3.2 \leqslant x < 3.6\) is obtained.
Carry out the goodness of fit test at the 10\% significance level.