Consider independent observations $(r_1,y_1),$dots$(r_n,y_n),$ from the model $Y_{i}Bin(r_i,p)$ for $i=1,$dots,$n,$ where the $r_i$ are fixed constants. Using likelihood $L(p)$ and $log-$likelihood $l(p)$ as appropriate, compute the following items.

Derive the maximum likelihood estimate hat$(p).$

Write the second derivative of $log-$likelihood $l(p).$

Give an expression of the approximated asymptotic standard error of hat$(p)$ by plugging in the estimate hat$(p).$ To this end, estimate the Fisher Information Matrix by hat$(V)=-\frac{de{l}^2}{del{p}^2}l(p){|}_p||=$hat$(p)$ and then s$.$e$.(hat(p))=\sqrt[\phantom{2}]{\text{h}\text{a}\text{t}(V{)}^{-1}}.$

Consider data $(15,11),(20,14),(15,9),(10,7),(25,17),(15,12),(10,8).$ Using your formul$,$ compute and write numerical estimates hat$(p),$ s$.$e$.(hat(p))$ and give a $95\%$ confidence interval for $p$ using the normal approximation.

Note: To answer this question you will work by hand. Do not write in the textbox but upload a single page pdf image of your workings and results. For theoretical computations $($a$)$ to $($c$)$ you are expected to show your equations and developments in detail, simplifying as much as possible; for the numerical calculations in $($d$)$ you can use R but only write the requested results.