Manufacturer knows that their items have a normally distributed length, with a mean of 15.9 inches, and standard deviation of 4.5 inches. if 10 items are chosen at random, what is the probability that their mean length is less than 18.4 inches?

70.08%

Explanation:

Hi! First of all we define the equation to know 'z' value:

z = (X - μ) / σ

Where:

X = 18.4

μ = mean = 15.9

σ = standard deviation = 4.5 inches

Then, we replace:

z = (18.4 - 15.9) / 4.5

z = 0.55

Next step is to search 'z' value in a z-score chart like this one:

https://www.stat.tamu.edu/~lzhou/stat302/standardnormaltable.pdf

On the second page, in the z column, you will find the value 0.5 in the sixth row. Then you must move through that row, until you reach the value column '.05'. Finally, that value represents the area of the probability sought.

In this case, for z = 0.55, the area is 0.708

So the probability is that 70.08% shall be less that 18.4 inches.

Hope it helps! :)

at right angles to each other