1. Let X ∼ uniform(0, θ) and consider testing H0: θ = 1 vs. H1: θ > 1. We take one

observation, X, from this distribution and reject H0 if X > 0.9.

(a) Find α, the type I error probability.

(b) Suppose that θ = 1.1. Find β, the type II error probability.

2.LetX1,...,Xn ∼N(μ, σ 2 =4). To test H0: μ=10 vs H0: μ>10 we take a random

sample of size n = 16 and reject the null hypothesis if x > 14.

(a) Find α, the type I error probability.

(b) Suppose that μ = 11. Find β, the type II error probability.

3. A soft-drink machine at a steak house is regulated so that the amount of drink

dispensed is approximately normally distributed with a mean of 200 milliliters.

(a) State the null and alternative hypotheses if we wish to check if the machine is

calibrated correctly.

(b) If a sample of size n=25yields x =198.1 and s=5.8 what can we say about the

current state of the machine. Use α = 0.05. Assume the measurements to be

normally distributed.

4. The following data represent the running times of films produced by two motionpicture companies:

Test the hypothesis that the average running time of films produced by

company 2 exceeds the average run ning time of films produced by

company 1 by 10 minutes against the one-sided alternative that the

difference is less than 10 minutes. Use a 0.1 level of significance and

assume the distributions of times to be approximately normal with

equal variances.

5. In a controlled laboratory experiment, scientists at the University of Minnesota

discovered that 25% of a certain strain of rats subjected to a 20% coffee bean diet and

then force­fed a powerful cancer­causing chemical later developed cancerous tumors.

Would we have reason to believe that the proportion of rats developing tumors when

subjected to this diet has increased if the experiment were repeated and 16 of 48 rats

developed tumors? Use a 0.05 level of significance.

6. An urban community would like to show that the incidence of breast cancer is

higher in their area than in a nearby rural area. (PCB levels were found to be higher

in the soil of the urban community.) If it is found that 29 of 200 adult women in the

urban community have breast cancer and 17 of 150 adult women in the rural

community have breast cancer, can we conclude at the 0.05 level of significance that

breast cancer is more prevalent in the urban community?

7. A survey asked a random sample of students in the university about their smoking

habits and their exercise levels. The results are shown in the table below.

Test if smoking habits and exercise frequency are independent of each

other.