A manufacturing process is designed to produce bolts with a 0.5-inch diameter. Once each day, a random sample of 36 bolts is selected and the bolt diameters recorded. If the resulting sample mean is less than 0.490 inches or greater than 0.510 inches, the process is shut down for adjustment. The standard deviation for diameter is 0.02 inches. What is the probability that the manufacturing line will be shut down unnecessarily

Using the normal distribution and the central limit theorem, it is found that there is a 0.0026 = 0.26% probability that the manufacturing line will be shut down unnecessarily.

In a normal distribution with mean and standard deviation , the z-score of a measure X is given by:

In this problem:

The manufacturing line will be shut down unnecessarily if the sample mean is less than 0.490 inches or greater than 0.510 inches.

By the Central Limit Theorem:

has a p-value of 0.0013.

2 x 0.0013 = 0.0026.

0.0026 = 0.26% probability that the manufacturing line will be shut down unnecessarily.

To learn more about the normal distribution and the central limit theorem, you can take a look at link

Given Information:  

Population mean = μ = 0.5  

Population standard deviation = σ = 0.02

Sample size n = 36

Required Information:  

Probability that the manufacturing line will be shut down unnecessarily = ?  

P = 0.9975

Explanation:  

The sample mean will be same as population mean μs = 0.5

The sample standard deviation is given by

σs = σ/√n

σs = 0.02/√36

σs = 0.0033

Now we can calculate the probability that the process will shut down for adjustment.

P(x < 0.490) = (x - μs)/σs

P(x < 0.490) = (0.490 - 0.50)/0.0033

P(x < 0.490) = -3.03

The corresponding z-score from the z table is 0.00122

P(x > 0.510) = (x - μs)/σs

P(x > 0.510) = (0.510 - 0.50)/0.0033

P(x > 0.510) = 3.03

As you can see the distribution is symmetric, therefore,

P( 0.510 < x < 0.490) = 2*0.00122

P( 0.510 < x < 0.490) = 0.00244

Notice that it was asked in the question to find the probability that the manufacturing line will be shut down unnecessarily, which means

P = 1 - P( 0.510 < x < 0.490)

P = 1 - 0.00244

P = 0.9975

Therefore, the probability that the manufacturing line will be shut down unnecessarily is 0.9975.

step-by-step explanation:

describing distributions is one of the key skills you’ll need to earn a high score on the ap statistics exam. if you need proof of this, just flip through some past exam questions, which can be found at the collegeboard website. you’ll notice that the first free response question is almost always a question requiring you to look at a graph and describe it.

step-by-step explanation: just trying to get points sorry : (