STA 350 HW 1

Questions 1-6 are from the textbook. You are allowed to use R for any of these, but please plan on turning a hard copy of your solutions into class (handwritten or printed is OK)

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Question 1: 1.3

Question 2: 1.4

Question 3: 1.7. For Part (b), also compare the probability distribution given with a binomial. Identify why the distribution function given should be used instead of the binomial

Question 4: 1.12. Part (d) can be determined using prop.test(). It may be a fun exercise to see if you can write a function to do this without prop.test() as well, but this is optional.

• Correction: I originally suggested using binom.test() for this, though this instead gives yet another confidence interval.
• Read the documentation for ?prop.test() to see how the score should be implemented without a correction
• If you already did this with binom.test() that’s fine

Question 5: 1.13

Question 6: 1.15

Question 7:

• Part A Find the likelihood function associated with the Poisson distribution
• Part B Using this, confirm that the MLE for the Poisson is

\[ \hat{\lambda} = \frac{1}{n} \sum_{i=1}^n x_i \]

Bonus Exercise

For this exercise, we will investigate comparative coverage of the Wald and Score confidence intervals for varying levels of \(\pi\) using simulation. Specifically, for a nominal 95% confidence interval, we want to estimate the proportion of repeated samples for which the derived confidence interval actually contains the true value of \(\pi\).

The simulation should use the following settings:

• Confidence intervals should have nominal coverage of 95%
• Each simulation should draw from a binomial random variable with \(n = 20\)
• Coverage should be evaluated at at least 50 separate values of \(\pi\) between 0.01 and 0.99
• For each value of \(\pi\), simulate \(N = 1000\) independent samples

The final product should be a ggplot with \(\pi\) on the x-axis and the estimated coverage probability on y-axis, with separate lines for the Score and Wald intervals. Include a horizontal reference line at \(y = 0.95\). Describe the resulting plot in 2-3 sentences.

Useful functions, though you may not need all (see e.g., ?rbinom() for use of each):

• rbinom()
• seq()
• replicate()
• mean(c(TRUE, TRUE, FALSE)) <- useful trick to know
• sapply()
• prop.test(x, n, correct = FALSE)$conf.int to get Score interval