$(13$ pts$)$ Consider the following frequency table of observations on the random variable $x$ :

a$)$ Based on these $50$ observations, is a binomial distribution with $n=6$ and $p=0\mathrm{.}25$ an

appropriate model? Perform a goodness$-$of$-$fit hypothesis test with $=0\mathrm{.}05.(8pts)$

b$)$ Calculate the p$-$value for this test, and explain its meaning in plain English. $(2$ pts$)$

c$)$ Still assuming that $n=6,$ use the method of moments to estimate $p.$ That is$,$ use $\stackrel{}{x}$ to

estimate $E[x],$ and use that to find an estimator for hat$(p).(1$ pt$)$

d$)$ If we were conducting the goodness$-$of$-$fit test using the estimated hat$(p),$ how would the

degrees of freedom for the test change? What would the new critical value for the test

be$?(2pts$