Respuesta :
Answer:
0.8185
Step-by-step explanation:
To solve this question, we have to understand the normal probability ditribution and the central limit theorem.
Normal probability distribution:
Problems of normally distributed samples can be 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 random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], 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]
In this problem, we have that:
[tex]\mu = 1.01, \sigma = 0.003, n = 9, s = \frac{0.003}{\sqrt{9}} = 0.001[/tex]
Find the probability that a random sample of n = 9 sections of pipe will have a sample mean diameter greater than 1.009 inch and less than 1.012 inch.
This is the pvalue of Z when X = 1.012 subtracted by the pvalue of Z when X = 1.009. So
X = 1.012
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central limit theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{1.012 - 1.01}{0.001}[/tex]
[tex]Z = 2[/tex]
[tex]Z = 2[/tex] has a pvalue of 0.9772
X = 1.009
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{1.009 - 1.01}{0.001}[/tex]
[tex]Z = -1[/tex]
[tex]Z = -1[/tex] has a pvalue of 0.1587
0.9772 - 0.1587 = 0.8185
0.8185 = 81.85% probability that a random sample of n = 9 sections of pipe will have a sample mean diameter greater than 1.009 inch and less than 1.012 inch.
