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  1. 18 points Determine which probability distribution would be most appropriate for modeling the quantity of interest: Bernoulli, Binomial, Normal, or Poisson. No explanation is required (note, it may be the case that not all assumptions are satisfied for your distribution of choice; that’s ok - for here, choose the most appropriate choice).
  1. 14 points Air pollution may cause lung deficiencies and development of diseases such as emphysema or COPD. A researcher is interested in the effect of air pollution on mice, and conducts an experiment whereby the researcher exposes mice to high levels of air exhaust for six hours a day for three weeks, and at the end of the three week period, evaluate whether the mice have developed deficiencies in lung function. In a sample of the mice used, 25 out of 100 developed such deficiencies. The researcher is interested in the number of mice that must be sampled before he obtains ten abnormal mice for use in further studies and proposes using a binomial distribution to model this quantity. Evaluate whether the assumptions for a binomial distribution are satisfied.
  2. 14 points Vaccination is one of the most effective public health tools ever created to control the spread of infectious, communicable diseases, but unfortunately recently there has been some pushback from the public against mandatory childhood vaccinations. Due to vaccine hesitancy, formerly well-controlled diseases such as measles have made a comeback in certain communities due to lack of vaccination and herd immunity. A researcher is interested in the number of children who develop measles across various school districts in the United States and proposes using a Poisson distribution to model this quantity. Evaluate whether the assumptions for a Poisson distribution are satisfied.
  3. 14 points A team of scientific researchers in Antarctica is investigating the number of eggs laid in a given season to Adelie penguin breeding pairs. In their initial survey of 100 randomly chosen breeding pairs, they found a mean of 2 eggs (standard deviation 0.3 eggs). Explain why it would or would not be appropriate to use a Poisson distribution to model this quantity.
  4. 15 points Assume that blood levels of vitamin B9 (in ng/mL) follow a normal distribution with mean 400 ng/mL and standard deviation of 100 ng/mL and blood levels of vitamin D in (ng/mL) follow a normal distribution with mean 50 ng/mL and standard deviation 10 ng/mL. Assume that blood levels of vitamins D and B9 are independent. Is it more likely that a randomly selected individual will have blood vitamin D concentration under 42 ng/mL or a blood vitamin B9 concentration above 520 ng/mL? If we cannot say using the information provided, please state why.
  5. 25 points FEV1, or forced expiratory volume in one second, is the volume of air that can be forcibly expelled in one second after taking a full breath. Suppose in a population of non-smoking 50-year old men, FEV1 is normally distributed with mean 4.0 liters and standard deviation 0.5 liters, and that in a population of 50-year old male smokers, FEV1 is normally distributed with mean 3.5 liters and standard deviation 0.5 liters. Let an FEV1 of 2.5 or less be defined to be evidence of lung function impairment (e.g., breathlessness when walking, etc.). Suppose that 20% of men in this population smoke, and suppose you take a random sample of 50 men. What is the expected number of men in this sample with impaired lung function?

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