Question

1. Why does the chi-square statistic tend to get larger as the degrees of freedom increase

2. Judy drew two bell-shaped distributions for her hypothesis test. One for the null hypothesis (the left one) and one for the actual truth (the right one). They overlap at a certain area. Her goal is to increase the area of the power. If she could move the distributions, would she move them closer together or farther apart to increase the power?

thanks

Answer 1

1.

The chi-square statistic with degree of freedom k is sum of squares of k standard normal random variables. The sum of squares of k standard normal random variables will increase with the value of k. Thus, the chi-square statistic tend to get larger as the degrees of freedom increase.

2.

The overlap area of null and alternative hypothesis determines the type II error (probability of fail to reject the null hypothesis given null hypothesis is false). If she could move the distributions farther apart, the overlap area will decrease and the type II error will decrease. The power of the test (probability to reject the null hypothesis given null hypothesis is false) is 1 - Type II error. Thus, as type II error will decrease, power increases. Thus, to increase the power, she could move the distributions farther apart.