Respuesta :
Answer:
0.0026 = 0.26% probability that the manufacturing line will be shut down unnecessarily
Step-by-step explanation:
We need to understand the normal probability distribution and the central limit theorem to solve this question.
Normal probability distribution
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score 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. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this problem, we have that:
[tex]\mu = 0.75, \sigma = 0.04, n = 36, s = \frac{0.04}{\sqrt{36}} = 0.0067[/tex]
What is the probability that the manufacturing line will be shut down unnecessarily?
Less than 0.73 or more than 0.77.
Less than 0.73
pvalue of Z when X = 0.73
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.73 - 0.75}{0.0067}[/tex]
[tex]Z = -3[/tex]
[tex]Z = -3[/tex] has a pvalue of 0.0013
More than 0.77
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.77 - 0.75}{0.0067}[/tex]
[tex]Z = 3[/tex]
[tex]Z = 3[/tex] has a pvalue of 0.9987
1 - 0.9987 = 0.0013
2*0.0013 = 0.0026
0.0026 = 0.26% probability that the manufacturing line will be shut down unnecessarily
