1. A company manufactures electronic sensors. A sample of 36 sensors is found to have an average lifetime of 352 days and a sample standard deviation of 20.0 days. The company advertises that the lifetime is one year, which for our purposes, take to be 365.25 days. Taking the company’s claim as the null hypothesis, using an α of 0.01, does the sample data support the alternate hypothesis that the lifetime is less than one year?2. Consider the following hypothesis test: H0 : µ = 3.2 V, H1 : µ 3.2 V. (a) Assume that the data is known to be Gaussian with σ = 0.32 V. A sample of n = 10 is to be taken. Determine the threshold on ¯ X for the hypothesis test, for a probability of false alarm α = 0.05. (b) Now assume that, in reality, the population mean µ = 3.4 V. For the threshold found in (a), what is the probability that we will decide H1? This is called the “probability of detection”. What is the probability that we will decide H0? This is called the “probability of missed detection”. (c) Let the sample size n be increased to n 10, and the threshold recomputed to achieve the same false alarm probability. Will the threshold be higher, or lower than when n = 10? If, in reality, the population mean µ = 3.4 V, will the probability of detection increase when n 10, compared to when n = 10? 3. We take 61 samples of a random variable that is supposed to have a standard normal distribution. We will use the sample data to test whether or not the mean is zero and whether or not the standard deviation (or equivalently, the variance) is equal to 1. (a) First, test the variance: H0 : σ2 = 1, H1 : σ2 6= 1. How high or how low does S2 need to be to reject the null hypothesis for an α = 0.05? (b) Second, test the mean: H0 : µ = 0, H1 : µ 6= 0. How high or how low does ¯ X need to be to reject the null hypothesis for an α = 0.05? (c) Assume that ¯ X and S2 are independent. Given H0 is true, what is the probability that we will see a false alarm in either the test on the variance or the test on the mean?
