The inside diameter of metal washers produced by a company are normally distributed with a mean of .5 inches and a standdrd deviation of .01 inches. Of all washers produced .8% of the washers are rejected because they are too small for a bolt used to test them. What is the diameter of the bolt used for the test?
Given:
μ=0.5"
σ=0.01"
Rejection rate = 0.8% = 0.008
rejection criterion = z<Z
X=μ+Zσ
=0.5+0.01Z
From normal distribution tables
P(Z<-2.409)=0.008
so
X=0.5-0.01*2.409
=0.4758
Ans: washers with diameters less than 0.476 are rejected.
To find the diameter of the bolt used for the test, we can use the concept of z-scores and the standard normal distribution.
Let's assume that the diameter of the bolt follows a normal distribution with the same mean and standard deviation as the metal washers.
First, we need to find the z-score corresponding to the rejection point of 0.8%. This can be calculated using the formula: z = (x - μ) / σ, where x is the value we are interested in, μ is the mean, and σ is the standard deviation.
We want to find the z-score such that the area to the left of it is 0.8%. Using a standard normal distribution table or a statistical calculator, we can find that the z-score corresponding to the left-tail area of 0.8% is approximately -1.28.
Now we can use the z-score formula to find the diameter of the bolt: z = (x - μ) / σ. Plugging in the values we know, -1.28 = (x - 0.5) / 0.01.
Solving for x, we get:
-1.28 = (x - 0.5) / 0.01
-1.28 * 0.01 = x - 0.5
-0.0128 = x - 0.5
x = 0.5 - 0.0128
x ≈ 0.4872
Therefore, the diameter of the bolt used for the test is approximately 0.4872 inches.