1. Let X ∼ uniform(0, θ) and consider testing H0: θ = 1 vs. H1: θ > 1. We take oneobservation, 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 randomsample 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 drinkdispensed 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 iscalibrated correctly.(b) If a sample of size n=25yields x =198.1 and s=5.8 what can we say about thecurrent state of the machine. Use α = 0.05. Assume the measurements to benormally 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 bycompany 2 exceeds the average run ning time of films produced bycompany 1 by 10 minutes against the one-sided alternative that thedifference is less than 10 minutes. Use a 0.1 level of significance andassume the distributions of times to be approximately normal withequal 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 ishigher in their area than in a nearby rural area. (PCB levels were found to be higherin the soil of the urban community.) If it is found that 29 of 200 adult women in theurban community have breast cancer and 17 of 150 adult women in the ruralcommunity have breast cancer, can we conclude at the 0.05 level of significance thatbreast cancer is more prevalent in the urban community?
7. A survey asked a random sample of students in the university about their smokinghabits and their exercise levels. The results are shown in the table below.
Test if smoking habits and exercise frequency are independent of eachother.