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.