A certain type of flashlight is sold with the four batteries included. A random sample of 150 flashlights is obtained, and the number of defective batteries in each is determined, resulting in the following data:

Let X be the number of defective batteries in a randomly selected flashlight. Test the null hypothesis that the distribution of X is Bin(4, ). That is, with pi
P(i defectives), test H0: pi
  i

(1  )4i i
0, 1, 2, 3, 4

[Hint: To obtain the mle of , write the likelihood (the function to be maximized) as u

(1  )v

, where the exponents u and v are linear functions of the cell counts. Then take the natural log, differentiate with respect to , equate the result to 0, and solve for

ˆ.]