Chapter 14.2, Problem 15E

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:

Number Defective	0	1	2	3	4
Frequency	26	51	47	16	10

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

      $\begin{array}{cc}{H}_0:p_i=\left(\begin{array}{c}4\\ i\end{array}\right){\theta }^i{(1-\theta )}^{4-i}& i=0,1,2,3,4\end{array}$

[Hint: To obtain the mle of θ, write the likelihood (the function to be maximized) as θ^u(1 – θ)^v, where the exponents u and v are linear functions of the cell counts. Then take the natural log, differentiate with respect to θ, equate the result to 0, and solve for $\hat{\theta }$.]