A statistician is modelling the proportion of allotted time students spend working on a certain

aptitude test. Suppose the random variable X denotes the proportion of allotted time that a

randomly selected student spends working on the aptitude test and the probability density

function of $x$ is given by

$f(x,)=(+1)x^{},0x1$

where

A random sample of $10$ students yields the following data.

a$)$ Use the method of moments to derive an estimator of the parameter $,$ by

finding $E(x),$ and then compute the estimate for this data.

b$)$ Obtain the maximum likelihood estimator of $,$ and then compute the MLE for

this given data. Calculate the standard error of your MLE and provide an

approximate $95\%$ confidence interval for $$ and comment on results. You must

show all your working and the reasons for the steps involved.