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Project 2: Simulation and p value

STAT35000, Summer 2017, Section 11872

Due: June 12 (Monday), 2017

Abstract

This is the 2nd R project. This problem is about sampling distribution by using R and calculation of the so called p value.

For the Bernoulli distribution X ∼ Ber(p) = Bin(1,p), figure out the follow- ing problems:

1. Sample 50 points from X ∼ Ber(p0 = 0.2).

2. Calculate the sample proportion p̂∗ of “1”s shown in the above sample you obtained.

3. Do you think it’s reasonably good enough? Explain why.

4. Pretend that you forgot the probability of success p0 you used to generate the above sample of size 50. Your guess now is 0.4. And you want to figure out whether it is 0.4. Here are two ways for your choice, and which one do you think is more reasonable?

(a) Compare the sample proportion you got in above part 2 with your guess 0.4. If they are reasonably close, you probably will adopt your guess 0.4. Think about how could you judge the closeness.

(b) Here is the other procedure. The logic behind this is that if p0 = 0.4, what will happen? If something strange happened, then you should doubt your guess; if everything happend is reasonable under p0 = 0.4, then it seems no reason for you to reject your guess. Here is the implementation. Use p0 = 0.4 to generate N = 10000 samples from X ∼ Ber(p0 = 0.4) with sample size n = 50. And calculate the sample proportion for each of these N = 10000 samples, denoted by {p̂k50,k = 1, 2 · · · ,N = 10000}. And then plot the histogram of {p̂k50,k = 1, 2 · · · ,N = 10000} to see the distribution of the random variable sample proportion p̂50 under the

1

2

assumption that the true p0 = 0.4. And if the observed p̂ ∗ is not in the

extreme region of the distribution, you probably will adopt your guess 0.4. In particular, calculate the probability that p̂50 < p̂

∗ through the frequency of {p̂k50 < p̂∗,k = 1, 2 · · · ,N = 10000}. This probability is actually related with the important concept in Statistics, p value!

5. For the above part (b), we are actually using simulation to approximate the probability of p̂50 =

X1+X2+···+X50 50

< p̂∗ given that {Xi, i = 1, 2, · · · , 50} are independent and identically distributed as X ∼ Ber(p0 = 0.4). Could you figure the probability out exactly without any approximation? What is the exact probability?

6. For the above probability, we are actually also be able to approximate it without simulation. Remember by the central limit theorem for n = 50 > 30,

p̂50 = X1 + X2 + · · · + X50

50

approximate∼ N(µ,σ2/50)

with µ = E(X) and σ2 = V ar(X) for X ∼ Ber(p0 = 0.4). Use this approximation fact, please calculate the probability.

7. Compare the p values you obtained by the above three ways (simulation approximation, exact, CLT approximation), you should expect to see that the CLT approximation is as good as the simulation approximation. There is some empirical continuity correction about this CLT approximation. Please check out the online material https://people.richland.edu/james/ lecture/m170/ch07-bin.html to figure out how to conduct the correction to make the approximation better. Calculate the corrected probability to earn 10 bonus points.