Given is a test with three Rasch items. The item parameters are b1 = −0.5, b2 = 0.0, and b3 = 0.5. A person has answered items 1 and 2 correctly, and item 3 incorrectly. Compute the likelihood for θ = −1.0, −0.5, 0.0, 0.5, 1.0.

a. Consider the four intervals defined by the five values of θ

for which the likelihood has been computed. In which interval lies the maximum likelihood estimate of θ?

b. Assume that we have a population distribution with five latent classes: θ1 = −1.0, θ2 = −0.5, θ3 = 0.0, θ4 = 0.5, θ5 =

1.0. Also assume that these latent classes have the same relative frequencies: g(θk) = 0.2 for k = 1,…, 5. Compute the EAP estimate of θ.