A certain type of flashlight is sold with the four batteries included. A random sample o...

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, θ ). That is, with p_i = P(i defectives), test

[Hint: To obtain the mle of θ , write the likelihood (the function to be maximized) as , where the exponents u and v are linear functions of the cell counts. Then take the natural log, differentiate with respect to u, equate the result to 0, and solve for ]