I have information of the order in which students were classified in regard to their scores in a SAT test. I know the distribution of scores for each student is uniform with support [a,b]. I also know that:
- 17 students scored less than 3
- 13 students scored between 3 and 5
- 58 students scored more than 5
How can I form reasonable estimates for a and b?
Thanks
Assume that $u$ students scored in $(x-r,x)$, $v$ students scored in $(x,x+s)$ and $w$ students scored in $(x+s,x+s+t)$, where $(u,v,w,x,s)$ are known and $(r,t)$ are unknown. The likelihood is $$ L(r,t)={u+v+w\choose u,v,w}\left(\frac{r}{r+s+t}\right)^u\left(\frac{s}{r+s+t}\right)^v\left(\frac{t}{r+s+t}\right)^w. $$ Thus, $L(r,t)$ is maximal when the partial derivatives of $L$ with respect to $r$ and $t$ are zero, that is, when $$ \frac{u}{r}=\frac{u+v+w}{r+s+t}=\frac{v}{t}. $$ Solving these yields $$ r=s\frac{u}v,\qquad t=s\frac{w}v, $$ a result which common sense could suggest directly. In the notations of the problem, $a=x-r$ and $b=x+s+t$ hence $a=3-\frac{17}{13}\cdot2$ and $b=5+\frac{58}{13}\cdot2$.