Respuesta :
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 = ?
Answer:
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.
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.
Normal Probability Distribution
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- It measures how many standard deviations the measure is from the mean.
- After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.
- By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
In this problem:
- The mean is of 0.5 inches, hence [tex]\mu = 0.5[/tex].
- The standard deviation is of 0.02 inches, hence [tex]\sigma = 0.02[/tex].
- A sample of 36 bolts is taken, hence [tex]n = 36, s = \frac{0.02}{\sqrt{36}} = 0.0033[/tex]
The manufacturing line will be shut down unnecessarily if the sample mean is less than 0.490 inches or greater than 0.510 inches.
- The normal distribution is symmetric, which means that these probabilities are equal, hence, we find one of them and multiply by 2.
- The probability that the resulting sample mean is less than 0.490 inches is the p-value of Z when X = 0.49, then:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem:
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.49 - 0.5}{0.0033}[/tex]
[tex]Z = -3[/tex]
[tex]Z = -3[/tex] 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 https://brainly.com/question/24663213
