Suppose that in a sequence of n Bernoulli trials, the probability p of success on each trial is unknown. Suppose also that p0 is a given number in the interval (0, 1), and it is desired to test the following hypotheses:
H0: p = p0,
H1: p = p0.
Let Xn denote the proportion of successes in the n trials, and suppose that the given hypotheses are to be tested by using a χ2 test of goodness-of-fit.
a. Show that the statistic Q defined by Eq. (10.1.2) can be written in the form

b. Assuming that H0 is true, prove that as n†’ˆž, the c.d.f. of Q converges to the c.d.f. of the χ2 distribution with one degree of freedom. Hint: Show that Q = Z2, where it is known from the central limit theorem that Z is a random variable whose c.d.f. converges to the c.d.f. of the standard normal distribution.

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Po(1- Po)