### 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 $$\mathcal{N}(2, 10^2)$$).

### Problem 1(a)

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

#### Solution

pnorm(q = 3.1, mean = 2, sd = 10) - pnorm(q = 2.1, mean = 2, sd = 10)
## [1] 0.03980596

Learning goal: compute probabilities of intervals

### Problem 1(b)

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

#### Solution

The idea here is that we can “shift” and “shrink” X using $$(X-2)/10$$ so that now $$(X-2)/10 \sim \mathcal{N}(0, 1)$$

pnorm(q = (3.1-2)/10, mean = 0, sd = 1) - pnorm(q = (2.1-2)/10, mean = 0, sd = 1)
## [1] 0.03980596

Learning goal: transform normal random variables to be $$\mathcal{N}(0, 1)$$

### Problem 1(c)

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

#### Solution

x <- rnorm(n = 100000, mean = 2, sd = 10)
mean((2.1 < x) & (x < 3.1))
## [1] 0.03982

Learning goal: compute probabilities via simulation. Understand the connection between samples from a distribution and the cumulative probability function.

### 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.

Learning goal: map a word problem to a cumulative probability computation, use the normal approximation to the binomial distribution.

#### Solution

1 - pbinom(q = 78, size = 100, prob = .65)
## [1] 0.001686446

Another option is to use the lower.tail argument, but that is not preferred right now

pbinom(q = 78, size = 100, prob = .65, lower.tail = F)
## [1] 0.001686446

Learning goal: map a word problem to a cumulative probability computation.

#### 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 $$n\times prob\times (1-prob)$$).

#### Solution

1 - pnorm(q = 78, mean = 65, sd = sqrt(.65*.35*100))
## [1] 0.003209814

(Note: we are not requiring trying to use a continuity correction. To match the answer to 2(b), we’d need q = 78.9)

Another option (dispreferred):

pnorm(q = 78, mean = 65, sd = sqrt(.65*.35*100), lower.tail = F)
## [1] 0.003209814

Learning goal: use the normal approximation to the binomial. Recognize the consequences of not using continuity correction.

### 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?

#### Solution

The null hypothesis here is that $$P(Stanford) = 0.5$$

The p-value here is P(n.Stanford >= 60 or n.Stanford <= 40). We can compute that using

pbinom(q = 40, size = 100, prob = 0.5) + (1 - pbinom(q = 59, size = 100, prob = 0.5))
## [1] 0.05688793

We would see a value that’s as extreme as what we’re seeing 5.6% of the time. This suggests that the data we have is consistent with Princeton students thinking that Harvard and Stanford are equally bad online institutions of higher learning.

### Problem 4

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

#### Solution

x <- rnorm(n = 100000, mean = 0, sd = 1)
# Now, 65 + x*sqrt(.65*.35*100) ~ N(65, sqrt(.65*.35*100)^2)
y <- 65 + x*sqrt(.65*.35*100)
mean(y > 78)
## [1] 0.00317

Learning goal: compute probability via simulation; flexibly apply variable transformation