Problem 1 (20 pts). Normal distributions are used to approximate the sampling distribution of the sample proportion pˆ when sample size is large (i.e., when np ≥ 10, n(1 − p) ≥ 10). Similarly, we can use a normal distribution to approximate a Binomial distribution when sample size is large, as shown in the lecture notes. However, the accuracies of these approximations are unclear (i.e., how close are the approximate answers to the exact answers if exact answers are available)? Here, you are asked to evaluate the accuracies of the normal approximations in several situations, based on a random variable X which follows the Binomial distribution B(n,p). You can easily obtain exact binomial probabilities from software R with command “pbinom” (some books also give exact binomial probability table. Calculating the exact binomial probabilities by hand is also feasible, though a bit tedious). For the following questions, you can obtain exact binomial probabilities using any of these approaches.

(a) (12 pts). Compute the following probabilities using both the Binomial distribution X ∼ B(n, p) (which gives an exact answer) and its normal approximation X ∼ N (np, np(1 − p)) (which gives an approximate answer), and compare the two answers to see if they are close or not:

(i) choose n = 10 and p = 0.2, and then compute P (X ≤ 3) using both methods; (ii) choose n = 10 and p = 0.4, and then compute P (X ≤ 3) using both methods; (iii) choose n = 50 and p = 0.2, and then compute P (X ≤ 8) using both methods; (iv) choose n = 50 and p = 0.4, and then compute P (X ≤ 8) using both methods;

Summarize the above results in a table and state your conclusions in no more than three sentences (i.e., in what cases the normal approximations seem most accurate – the exact answers and approx- imate answers are closest). (b) (8 pts). The sample proportion pˆ can be defined as pˆ = X/n, where X ∼ B(n, p). For the four cases (i)–(iv) in question (a), compute the probabilities P(pˆ ≤ 0.3) using normal approximations (i.e., pˆ ∼ N (p, p(1 − p)/n)). Intuitively (or based on what you have observed in (a)), which ap- proximation(s) do you think may be the most accurate one? (Note: the sample proportion pˆ has no exact distribution available, so we have to use normal approximations.)

(a) Exact and normal approximationsR-Output1) Exact> pbinom(3,10,0.2) [1] 0.8791261 > pbinom(3,10,0.4) [1] 0.3822806 > pbinom(8,50,0.2) [1] 0.3073316 > pbinom(8,50,0.4) [1]…