15. A certain type of ashlight is sold with the four batteries included. A random sample of 150 ashlights 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 ashlight. Test the null hypothesis

 n # 1 pˆ 2x1 # qˆ

pˆ

pˆ 1gxi  n2/gxi pˆ

pgxin # qn 1px11 # q2 # . . . # 1pxn1 # q2 that the distribution of X is Bin(4, u). That is, with pi  P(i defectives), test

[Hint: To obtain the mle of u, write the likelihood

(the function to be maximized) as uu(1  u)v, 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 .]