### Problem 1

Suppose that $$X\sim\mathcal{N}(2, 10^2)$$. We sample the variable $$X$$ once (i.e., we obtain a sample from the distribution $$\mathcal{N}(2, 10^2)$$). In this problem, you will be computing the same quantity in four different ways. You should expect to get roughly the same answer every time.

### Problem 1(a)

Write R code to obtain $$P(2.1 < X < 3.1)$$. Use pnorm.

### Problem 1(b)

Write R code to obtain $$P(2.1 < X < 3.1)$$. Use pnorm(..., ,mean = 0, sd = 1).

### Problem 1(c)

Write R code to obtain $$P(2.1 < X < 3.1)$$. Use rnorm. (And not pnorm.)

### Problem 1(d)

Write R code to obtain $$P(2.1 < X < 3.1)$$. Use rnorm(..., mean = 0, sd = 1)

### Problem 2

Suppose 65% of Princeton students like Wawa better than World Coffee. We selected a random sample of 100 students, and asked them which they prefer. What is the probability that more than 78 students said “Wawa”?

#### Problem 2(a)

Answer the question using pbinom.

#### Problem 2(b)

Answer the question using pnorm. Use the normal approximation to the Binomial distribution (recall: the mean is $$n\times prob$$ and the variance is $$n\times prob\times (1-prob)$$).

(Note: you shouldn’t expect an exact match between 2(a) and 2(b) because of the lack of continnuity correction. You can try obtaining an exact match by varying the value of the q).

### Problem 3

Suppose 100 Princeton students we asked whether Harvard or Stanford is the worse online institution of higher learning. 60 students said that Stanford is worse. Compute the p-value for the null hypothesis that Princeton students think that Harvard and Stanford are equally bad, on average. What can you conclude?

### Problem 4

Answer Problem 2 using only rnorm(..., mean = 0, sd = 1)