Question

Say you think that an output will be normally distributed with mean 1 and variance 1 You use the test statistic r- 1 a) At what values of x would you reject the null if your a value is set to 0.05. Hint: there are two values xŻ and xk such that if x x; or if x x, we would reject. b) Say that in actuality, z is drawn from an exponential distribution with mean 1. Thus the null hypothesis is false. What is the type II error rate of your test described in (a)?

Answer 1

a)

Critical z value for $\alpha$ = 0.05 is $\pm$1.96

To reject null hypothesis, test statistic is less than -1.96 or greater than 1.96

x - 1 < -1.96 or x - 1 > 1.96

=> x < -0.96 or x > 2.96

b)

X ~ Exp(1)

Type II error rate = P(Do not reject H0 | H0 is false)

= P(-0.96 < x < 2.96)

= P(0 < x < 2.96) as X > 0

= 1 - exp(-2.96)

= 0.9482