Page 78 - Applied Statistics with R

78 CHAPTER 5. PROBABILITY AND STATISTICS IN R

Thus,

(0 < < 2) = ( < 2) − ( < 0).

This can then be calculated using R without a need to first standardize, or use
a table.

pnorm(2, mean = 1, sd = sqrt(0.32)) - pnorm(0, mean = 1, sd = sqrt(0.32))

## [1] 0.9229001

An alternative approach, would be to simulate a large number of observations
of then use the empirical distribution to calculate the probability.
Our strategy will be to repeatedly:

2
• Generate a sample of 25 random observations from ( = 6, = 4).
1

Call the mean of this sample ̄ .
1
2
• Generate a sample of 25 random observations from ( = 5, = 4).
1
Call the mean of this sample ̄ .
2
• Calculate the differences of the means, = ̄ 1 − ̄ .

2
We will repeat the process a large number of times. Then we will use the
distribution of the simulated observations of as an estimate for the true

distribution of .
set.seed(42)
num_samples = 10000
differences = rep(0, num_samples)
Before starting our for loop to perform the operation, we set a seed for repro-
ducibility, create and set a variable num_samples which will define the number
of repetitions, and lastly create a variables differences which will store the
simulate values, .

By using set.seed() we can reproduce the random results of rnorm() each
time starting from that line.
for (s in 1:num_samples) {
x1 = rnorm(n = 25, mean = 6, sd = 2)
x2 = rnorm(n = 25, mean = 5, sd = 2)
differences[s] = mean(x1) - mean(x2)
}