Question

PVC pipe is manufactured with a mean diameter of 1.01 inch and a standard deviation of 0.003 inch. The diameters are known to be normally distributed. 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. (Express the final answer to four decimal places (e.g. 0.9876).)

Answer 1

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 $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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 $\mu$ and standard deviation $\sigma$, the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 1.01, \sigma = 0.003, n = 9, s = \frac{0.003}{\sqrt{9}} = 0.001$

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

$Z = \frac{X - \mu}{\sigma}$

By the Central limit theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{1.012 - 1.01}{0.001}$

$Z = 2$

$Z = 2$ has a pvalue of 0.9772

X = 1.009

$Z = \frac{X - \mu}{s}$

$Z = \frac{1.009 - 1.01}{0.001}$

$Z = -1$

$Z = -1$ 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.