1) The Environmental Protection Agency (EPA) is charged with monitoring “acid rain," a term for the fall of water through an acidic atmosphere. Acidity is measured on the pH scale, where pure water...

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1) The Environmental Protection Agency (EPA) is charged with monitoring “acid rain," a term for the fall of water through an acidic atmosphere. Acidity is measured on the pH scale, where pure water has a pH of 7.0. Normal rain is slightly acidic because of atmospheric carbon dioxide, and has a pH of about 5.5 (lower pH indicates greater acidity). Suppose the EPA wishes to determine whether a particular area is subject to acid rain. Let ? denote the true average for pH in this area. (a) What is the appropriate null hypothesis in this study? (b) What is the appropriate alternative hypothesis in this study? (c) In the context of this study, describe a Type I and a Type II error. 2. Bats are nocturnal mammals, feeding on insects at night and sleeping during the day. Many species of bats use bridges as daytime sleeping places. Their choice of daytime position on the undersurface of a bridge appears to be non-random. One theory is that the bats choose locations that will keep them safer from predators. The beams that support a bridge create two kinds of spaces: wide (approximately 55 cm) and narrow (approximately 17 cm); biologists believe the narrow spaces provide more safety. Investigators studying the sleeping position choices of the Big-eared Bat (Corynorhinus rafinesquii) observed that 67 out of 102 of them were in narrow beam spaces. The narrow beam spaces accounted for approximately 46% of the available area under the bridges. Does this sample provide sufficient evidence that the bats prefer narrow over wide sleeping space? Use a significance level of 0.05 to test the appropriate hypothesis. Your answer should show assumptions, hypotheses, and conclusions. 3. The department of natural resources classifies a fish to be unsafe to eat if its polychlorinated biphenol (PCB) concentration exceeds 5 parts per billion (ppb). A random sample of 10 fish taken from a local lake resulted in the concentrations listed below. The population is known to have an approximately normal distribution. 2.6 3.7 3.8 4.5 4.8 4.9 5.1 5.4 6.6 6.9 At a 10% significance level, is there sufficient evidence to conclude that the mean PCB concentration for fish from this lake exceeds 5 ppb? Hint: ? is unknown. Your answer should show assumptions, hypotheses, and conclusions. 4. The mean salt content of a certain type of low-salt potato chip is supposed to be 2.0 mg. The salt content of these chips varies normally with standard deviation ? = 0.1 mg. From each batch produced, an inspector takes a sample of 50 chips and measures the salt content of each chip. The inspector rejects the entire batch if the sample mean salt content is significantly higher than 2.0 mg at the 5% significance level. For each question, select the best answer. (i) Suppose the mean salt content in a sample of 50 chips is found to be ?̅ = 2.03 mg. You determine that the ?-value is 0.017. What conclusion would you make? (a) The mean salt content is not significantly higher than 2.0 mg at the 5% significance level. (b) The batch should be discarded because the ?-value is not equal to ?. (c) The mean salt content is significantly higher than 2.0 mg at the 5% significance level. (d) More information is needed. Answer (ii) Which of the following decisions would be made due to a Type II error? (a) The batch is discarded when the true mean salt content is not significantly higher than 2.0 mg at the 5% significance level. (b) The batch is discarded when the true mean salt content is significantly higher than 2.0 mg at the 5% significance level. (c) The batch is not discarded when the true mean salt content is significantly higher than 2.0 mg at the 5% significance level. (d) The batch is not discarded when the true mean salt content is not significantly higher than 2.0 mg at the 5% significance level. Answer (iii) What is the smallest value of ?̅ that would lead to a decision to discard a batch of chips? (a) 2.1645 (b) 2.1960 (c) 2.0277 (d) 2.0233 Answer 5. For your initial post, explain why we want to design studies so that a Type I error is more serious than a Type II error.