The breaking strength (in pounds) of a certain new synthetic is normally distributed, with a mean of 145 and a variance of 25. The material is considered defective if the breaking strength is less than 134.5 pounds. What is the probability that a single, randomly selected piece of material will be defective

Standard deviation (sd) is the square root of the variance

so sd = √25 = 5

z- score = (x - mean)/sd
= (134.5-145)/5 = -2.1

now you go to your table or chart provided in your text , find -2.1 in your left column to see
.0179
so the prob is .0179

Many newer textbooks these days dealing with that topic don't even publish the tables any more, relying rather on "technology" to do the look-up

My favourite page for this is
http://davidmlane.com/normal.html
click on "below" and enter -2.1 to get the above result
The beauty of this site is that you don't even have to find the z-scores first,
in the mean, enter 145
in SD, enter 5
click on below, and enter 134.5
and behold.... .0179 !!!!!!!

Find the monthly payment for the loan. (Round your answer to the nearest cent.)

$800 loan for 12 months at 10%

To find the probability that a randomly selected piece of material will be defective, we need to calculate the area under the normal distribution curve to the left of the value 134.5.

First, we need to standardize the value 134.5 using the standard deviation (the square root of the variance). The standard deviation in this case is √25 = 5.

To standardize the value, we calculate the z-score using the formula:

z = (x - μ) / σ

where x is the value, μ is the mean, and σ is the standard deviation.

z = (134.5 - 145) / 5 = -2.1

Next, we need to find the area under the standard normal distribution curve to the left of z = -2.1. We can use a standard normal distribution table or a statistical calculator to find this area.

Using a standard normal distribution table, we look up the value for z = -2.1 and find that the area to the left of z = -2.1 is approximately 0.0179.

Therefore, the probability that a single, randomly selected piece of material will be defective is approximately 0.0179, or 1.79%.