If 3 items are chosen at random, what is the probability that their mean length is less than 7 inches?

Question

A manufacturer knows that their items have a normally distributed length, with a mean of 8.1 inches, and standard deviation of 2 inches.

If 3 items are chosen at random, what is the probability that their mean length is less than 7 inches?

Answer 1

Hi Kaitlyn,

This is another case of the classic basic stats equation z=x-mu/sigma, but with a twist. Because 3 items is a sample, not a population, you must first adjust your standard deviation to obtain a metric called the standard error using the following formula:

SE=s/sqrt(n) where:

SE=standard error, metric we are looking for

s=standard deviation=2

n=sample size=3

Thus:

SE=2/sqrt(3)

SE=1.15

Now, back to our classic equation:

z=x-mu/sigma--for samples, this becomes:

z-x-x-bar/se where:

x=7 in.

x-bar=mean=8.1 in.

se=standard error=1.15, calculated above

Thus:

z=(7-8.1)/1.15

z= -0.96

From z-table,

P(Z< -0.96)=P(X<7)=0.1685

P(X<7)=0.1685

I hope this helps. Good luck.