library(ggplot2)

4.39 Weights of pennies.- The distribution of weights of United States pennies is approximately normal with a mean of 2.5 grams and a standard deviation of 0.03 grams.

(a) What is the probability that a randomly chosen penny weighs less than 2.4 grams?

mean = 2.5
sd = 0.03
pnorm(2.4,mean = 2.5,sd = 0.03)
## [1] 0.0004290603

(b) Describe the sampling distribution of the mean weight of 10 randomly chosen pennies.

Assuming the 10 randomely choosen pennies are normally distributed.

n= 10
SampleD <- sd/(n^0.5)
SampleD
## [1] 0.009486833

sampling distribution of 10 samples is approximately normal with standard error = 0.009

(c) What is the probability that the mean weight of 10 pennies is less than 2.4 grams?

pnorm(2.4,mean = 2.5,sd = SampleD)
## [1] 2.797279e-26

The probability is very small tending towards zero.

(d) Sketch the two distributions (population and sampling) on the same scale.

normsample <- seq(mean - (4 * sd), mean + (4 * sd), length=15)
randomsample<- seq(mean - (4 * SampleD), mean + (4 * SampleD), length=15)
hnorm <- dnorm(normsample,mean,sd)
hrandom<- dnorm(randomsample,mean,SampleD)

plot(normsample, hnorm, type="l",col="blue", ylim=c(0,40))
lines(randomsample, hrandom, col="red")

# red = sampling,blue = population

(e) Could you estimate the probabilities from (a) and (c) if the weights of pennies had a skewed distribution?

We will not be able to get the estimate from (a) as (a) asks for probability less than 2.4.hence it is skewed to right and not normal.We can derive from (c) but sample size 10 is way to small to get any prediction.