7 A continuous random variable \(X\) has cumulative distribution function F given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < - 1 ,
\frac { 1 } { 2 } \left( x ^ { 3 } + 1 \right) & - 1 \leqslant x \leqslant 1 ,
1 & x > 1 . \end{cases}$$ Find \(\mathrm { P } \left( X \geqslant \frac { 3 } { 4 } \right)\), and state what can be deduced about the upper quartile of \(X\). Obtain the cumulative distribution function of \(Y\), where \(Y = X ^ { 2 }\). 8150 sheep, chosen from a large flock of sheep, were divided into two groups of 75 . Over a fixed period, one group had their grazing controlled and the other group grazed freely. The gains in weight, in kg, were recorded for each animal and the table below shows the sample means and the unbiased estimates of the population variances for the two samples. It is required to test whether the population mean for sheep having their grazing controlled exceeds the population mean for sheep grazing freely by less than 5 kg . State, giving a reason, if it is necessary for the validity of the test to assume that the two population variances are equal. Stating any other assumption, carry out the test at the 5\% significance level.