Question

A manufacturer knows that their items have a normally distributed length, with a mean of 10.8 inches, and standard deviation of 0.9 inches.If one item is chosen at random, what is the probability that it is less than 8.4 inches long?

Answer 1

We need to find the probability:

$P(X<8.4)$

where X is a normal random variable with mean 10.8 and standard deviation 0.9. To find this probability we need to use the z-score formula so we can use the standard normal distribution. The z-score is given by:

$z=\frac{x-\mu}{\sigma}$

where μ is the mean and σ is the standard deviation. In this case the z-score is given as:

$\begin{gathered} z=\frac{8.4-10.8}{0.9} \\ z=-2.67 \end{gathered}$

Then we have that:

$P(X<8.4)=P(z<-2.67)$

Looking for the probability on the right side of the previous expression in the standard table we have that:

$P(X\lt8.4)=P(z\lt-2.67)=0.0038$

Therefore, the probability of choosing an item with length less than 8.4 inches is 0.0038