ABSTRACT
To demonstrate this estimation procedure with a computationally simple example, suppose that we select a random sample of 10 students from the population of all students in the United States and record whether each student is proficient (a “success,” in the terminology of Chapter 2) or not proficient in mathematics. Since the proficiency outcome for each student is a Bernoulli trial, and there are n 10 such trials, the appropriate underlying
distribution for this process is the binomial. Recall (from Chapter 2) that the binomial probability of k successes in n independent “trials” is computed as
P Y k n
k k n k( ) ( ) .( )
¤
¦¥ ³
µ´ P P1 (3.1)
Using Equation 3.1, suppose that in our example 4 of the 10 students were proficient in mathematics. The probability would thus be computed by substituting n 10 and k 4 into Equation 3.1, so
P Y( ) ( ) .( )
¤
¦¥ ³
µ´ 4 10
4 14 10 4P P
By evaluating this probability for different values of Q, the maximum likelihood estimate is the value of Q at which the probability (likelihood) is highest (maximized). For example, if Q 0.3, the probability of 4 (out of the 10) students being proficient is
P Y( ) ( . ) ( . )
(
¦¥ ³
µ´
r r r r
4 10 4
0 3 1 0 3
10 9 8 7 6
r r r r r r r r r r r r r
5 4 3 2 1 4 3 2 1 6 5 4 3 2 1
0 3 0 74 6 )
( )( ) ( . ) ( . ) 210 0 0081 0 1176 0 20( . )( . ) . .
