Presentation for Class

Problem 4.39. Weight 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
#using the Z score formula
(2.4-mean)/sd
## [1] -3.333333
#we can assume ND so we can use pnorm to get the prob.
pnorm(2.4, mean = mean, sd = sd)
## [1] 0.0004290603

There is a 0.04% chance of a penny weighing less than 2.4 grams.

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

since we are now sampling from a nearly normal distribution, this means we are sampling from a sample distribution.

This means that our standard of error for the new sample is the population sigma divided by the square root of the sample size.

n=10
SDsample = sd/n^0.5
SDsample
## [1] 0.009486833

This means that we are also making the assumption that the sample of 10 pennies is also approximately normal with mean = the sample population mean and standard error as sigma / sqrt n N(mean, sd/n^0.5) or 0.009

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

in this case the Z score still uses the population mean but now uses the sigma of the population which is sd/sqrt n

(2.4-2.5)/SDsample
## [1] -10.54093
pnorm(2.4, mean = mean, sd = SDsample)
## [1] 2.797279e-26

the probability is so small it is virtually 0.

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

#plot 1 sample population
x <- seq(2.4,2.6, length=100)
y1 <- dnorm(x=x, mean=mean, sd=sd)
#plot 2 sample from population
y2 <- dnorm(x=x, mean=mean, sd = SDsample)

#plotted together
par(mfrow=c(1,1))
plot(x,y1, main ="N(mu = mean, sigma = sd)")+
plot(x,y2, main ="N(mu = mean, sigma = SD Sample, n=10)")

## numeric(0)

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

For a, we wouldn’t be able to since the population itself is skewed and couldn’t be normal. However for c, if it met the conditions for being nearly normal (IID, n>=30, and not STRONGLY skewed) then yes we would be able to estimate probbilities, however at a sample of n=10 it is not possible.