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1. (Hypothesis Testing) Coin flip: Let p be the probability of showing heads in one flip experiment. We wish to test Ho p = 0.5, against Ha p> 0.5. The test statistic 0 is Y the number of heads observed. Let's do 50 experiments and we observed Y = 30. Use a test level 0.05 to make a conclusion if the data provide sufficient evidence to support our research hypothesis Ha. 2. (Large-Sample Tests) Shear strength measurements derived from unconfined com- pression tests for two types of soils gave the results shown in the following table (measurements in tons per square foot), where or and T2 represent the standard de- viation of measured data. Do the soils appear to differ with respect to average shear strength, at the 1% significance level? Soil #1 Soil #2 721 35 n.2 40 y1 1.68 y/2 1.45 o1 0.26 T2 0.23 3. (Compute Type II Error) We have two different cakes (red and blue), and we want to check the weight of fat in them. Independent random samples of 55 red cakes and 55 blue cakes were collected, as shown in the table. Testing hypotheses: Ho 0 = 11 - 12 = Oo against Ha 0 = 11 - 12 > 0o. Red Blue 72.1 55 112 55 P1 = 4.0 g 12 3.8 g of = 0.36 a: 0.24 (a) Choose correct Oo and answer: do the red cakes contain more fat than the blue cakes, at the 5% significance level? 1 (b) Given a = 0.05, Oo = 0 and OF = 0.3. what's the probability of Type II error? (c) Given a = 0.05, B = 0.05, 00 = 0 and 0. = 0.3, find the sample size for the upper-tailed test. 4. (p-values) A pharmaceutical company conducted an experiment to compare the mean times (in days) necessary to recover from the effects and complications that follow the onset of the common cold. This experiment compared persons on a daily dose of 500 milligrams (mg) of vitamin C to those who were not given a vitamin supplement. For each treatment category, 55 adults were randomly selected, and the mean recovery times and standard deviations for the two groups were found to be as given in the accompanying table. Treatment No supplement 500mg Vitamin C 72.1 55 72.2 55 M 6.5 M2 5.8 01 2.9 o2 1.2 (a) Do the data indicate that the use of vitamin C reduces the mean time required to recover? Find the attained significance level. (b)What would the company conclude at the a = 0.05 level? 5. (Small Sample Tests) The price of a popular sneaker at a national chain store is $52. Jack bought six of the same sneaker at an online auction site for the following prices: Price: $44 $46 $49 $58 $43 $38 Assuming that the auction prices of sneakers are normally distributed, determine whether there is sufficient evidence in the sample, at the 5% level of significance, to conclude that the average price of the sneaker is less than $52 if purchased at an online auction. 2

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