Please show complete steps and explanation

1. Suppose x is a single observation on a random variable X Exp(). We wish to test the null hypothesis H0 : 1 = 100 against the alternative H1 : 1 > 100. We decide to reject H0 if x 460.

(i) Considering the possible values of , what are the sets S, and which define this hypothesis test?

(ii) What is the sample space A for this test, and what are the acceptance region A0 and the rejection region A1?

(iii) Calculate the probability of making a Type 1 error.

2. Suppose x is a single observation from a random variable X which is dis- tributed X Binomial(20, ). We wish to test H0 : = 0.5 against H1 : < 0.5). We decide to reject H0 if x 6.

(i) Considering the possible values of , what are the sets S, and which define this hypothesis test?

(ii) What is the sample space A for this test, and what are the acceptance region A0 and the rejection region A1?

(iii) Calculate the probability of making a Type 1 error.