19. Each headlight on an automobile undergoing an annual vehicle inspection can be focused either too high (H), too low (L), or properly (N). Checking the two headlights simultaneously (and not distinguishing between left and right) results in the six possible outcomes HH, LL, NN, HL, HN, and LN. If the probabilities

(population proportions) for the single headlight focus direction are P(H)  u1, P(L)  u2, and P(N)  1  u1  u2 and the two headlights are focused independently of one another, the probabilities of the six outcomes for a randomly selected car are the following:

p1  p2  p3  (1  u1  u2)2 p4  2u1u2 p5  2u1(1  u1  u2)

p6  2u2(1  u1  u2)

Use the accompanying data to test the null hypothesis where the pi (u1, u2) s are given previously.

Outcome HH LL NN HL HN LN Frequency 49 26 14 20 53 38

(Hint: Write the likelihood as a function of u1 and u2, take the natural log, then compute 0/0u1 and 0/0u2, equate them to 0, and solve for .)