Let X be the number of Chargers (NFL team in Los Angeles) fans observed in a random sample of n = 30 graduate students at Penn State’s University…

  1. Let X be the number of Chargers (NFL team in Los Angeles) fans observed in a random sample of n = 30 graduate students at Penn State’s University Park campus. Assume that X ∼ Bin(30; π). The true proportion π is unknown, but it is likely to be small (especially in comparison to the potential number of Pittsburgh Steelers fans).
  2. (a) Assume for now that π = 0.04. Find the probability that X = 0.
  3. (b) The classic (Wald) approximate 95% confidence interval for π is
  4. πˆ(1 − πˆ) n
  5. where πˆ = X/n. When does this interval become degenerate (i.e., when the lower and upper bounds are the same)? If the true π were actually 0.04, could this interval actually cover the true parameter 95% of the time? Why or why not? Hint: based on part (a), how often would the interval become degenerate at zero?
  6. (c) Suppose that we observe two Chargers fans in the sample. Plot the log-likelihood func- tion for π over a range of values from 0.01 to 0.20. Find the ML estimate, and calculate the approximate 95% confidence interval for π based on the formula above.
  7. (d) Now find an approximate 95% confidence interval based on the likelihood ratio method. That is, find the range of null values that the likelihood ratio test would fail to reject.
  8. (e) In their 1998 paper, Agresti and Coull propose using the Wald interval but with two pseudo successes and two pseudo failures first added to the sample. Compute the 95% interval based on this modification, and compare it with the other intervals calculated above.