Question

Heated-treated parts are specified to have a hardness of at least 45 (lower specification limit). Any parts that are below 45 are too soft and have to be heat treated again at great expense. The hardness of heat-treated parts actually produced follows a normal distribution with a mean of 48.5 with a standard deviation of 2.

Note: a sketch of the distribution helps the set up your calculation.
What is the probability that a part is below the minimum specified hardness?

Answer 1

P(X<45)= 0.0401

Explanation:

Hello!

The study variable is:

X: Hardness of heat-treated parts.

This variable has a normal distribution with μ= 48.5 and standard deviation δ= 2

You need to calculate the probability of the parts being below the minimum specified hardness. If the lowest specificated hardness of heat-treated parts is 45, then:

P(X<45)

To calculate this probability you have to use the standard normal distribution: Z=(X-μ)/δ~N(0;1)

Using the information of the mean and standard deviation of the population you have to standardize the value of X so that you can use the tabulated Z-distribution to reach the asked probability:

P(Z<(45-48.5)/2)= P(Z<-1.75)

Since the value is negative you have to look for the probability in the left Z-table. The integer and first decimal, -1.7* value is in the first column and the second integer is in the first row, *.*5. When you cross them you find the corresponding probability.

P(Z<-1.75)= 0.0401

(sketch of distribution and table are attached)

I hope it helps!