Exam 2011, question A7

Individual weights of a feral strain of cat are normally distributed with a mean of 2400 g and variance of 642 g².

Calculate the following probabilities:

1. A randomly selected cat weighs less than 2385 g.

meanCats <- 2400
varCats <- 642
sdCats <- sqrt(varCats)
pnorm(2385, meanCats, sdCats)

## [1] 0.2769244

2. Two randomly selected cats each weigh less than 2385 g.

pnorm(2385, meanCats, sdCats)^2

## [1] 0.0766871

3. The mean of a sample of 10 randomly selected cats is less than 2385 g.

mean10 <- meanCats
sd10 <- sdCats/sqrt(10)
pnorm(2385, mean10, sd10)

## [1] 0.03059796

4. The difference in weight between two randomly selected cats is greater than 23.85 g.

meanDiff <- 0
varDiff <- varCats * 2
pnorm(23.85, meanDiff, sqrt(varDiff), lower.tail = FALSE) * 2

## [1] 0.5056741

# Simulation
simSize <- 1e6
cat1 <- rnorm(simSize, meanCats, sdCats)
cat2 <- rnorm(simSize, meanCats, sdCats)
diffs <- cat2 - cat1
length(diffs[diffs > 23.85 | diffs < -23.85])/simSize

## [1] 0.505616