Question

In Example 4.30 explain why the power goes to 1 as n gets large. (Hint: Look at what happens to the numerator and denominator in the statistic , X) /VS'(n)/n as n gets large.) Example 4.30 For the null hypothesis Ho: μ-1 in Example 4.29, we can estimate the power of the test when, in fact, the X's have a normal distribution with μ-1.5 and standard deviation ơ- l. (This is H.) We randomly generated 1000 independent observations of the statistic t [X(10) 1/Vs'(10)/10 under the assumption that H, is true. For 433 out of the 1000 observations>.83 and, therefore, the es- timated power is 0.433. Thus, if H is true, we will only rejeci the null hypothesis H, approximately 43 percent of the time for a test at leve 0.10. To see what effect the sample size n has on the power of the test, we generated 1000 observations of (n 25) when H, is true and also 1000 observationsof(n 100) when H,is true (all X,'s were normal). The estimated powers were δ-, 0.796 and δ-0.999, respec- tively. It is not surprising that the power is apparently an increasing function of n, since we would expect to have a better estimate of the true value of u when n is large. INote that in the case of normal sampling and a known standard deviation, as in this example, the power of the test can actually be computed numerically, obviating the need for simu- lation as done here: see, for example, Devore (2008, pp. 302-303).]

Answer 1

when n increases ,

numerator remains roughly the same

whereas denominator decreases as   S(n) /sqrt(n) decreases

hence statistic increases

critical value will tend to 1.96 - same as normal distribution

TS > 1.96 as n goes to infinity