4.7 Show algebraically that when ei nui and k 2, we have Q X2 i1 (Fi ei)2 ei (F1

4.7 Show algebraically that when ei ¼ nui and k ¼ 2, we have Q ¼

X2 i¼1

(Fi  ei)2 ei ¼

(F1  nu1)2 nu1(1  u1)

so that when k ¼ 2, ffiffiffiffi pQ is the statistic commonly used for testing a hypothesis concerning the parameter of the binomial distribution for large samples. By the central-limit theorem, the distribution of ffiffiffiffi pQ approaches the standard normal distribution as n!1and the square of any standard normal variable is chi-square-distributed with 1 degree of freedom. Thus we have an entirely different argument for the distribution of Q when k ¼ 2.