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Chapter 6: Problem 45

A machine that cuts corks for wine bottles operates in such a way that the distribution of the diameter of the corks produced is well approximated by a normal distribution with mean \(3 \mathrm{~cm}\) and standard deviation \(0.1 \mathrm{~cm} .\) The specifications call for corks with diameters between 2.9 and \(3.1 \mathrm{~cm}\). A cork not meeting the specifications is considered defective. (A cork that is too small leaks and causes the wine to deteriorate; a cork that is too large doesn't fit in the bottle.) What proportion of corks produced by this machine are defective?

Short Answer

Expert verified
The proportion of corks produced by the machine that are defective is approximately 31.73%

Step by step solution

01

Standardize the values

According to the formula \(Z = \frac{(X - \mu)}{\sigma}\), where X is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation, standardize the lower limit (2.9cm) and the upper limit (3.1cm). This will give the standard deviations away from the mean.
02

Calculate the Z-scores

Use the formula from the previous step, substituting 2.9cm for \(X\) for the lower limit, and 3.1cm for \(X\) for the upper limit. This yields \(Z_{lower} = \frac{(2.9 - 3)}{0.1} = -1\) and \(Z_{upper}= \frac{(3.1 - 3)}{0.1} = 1\).
03

Calculate the proportion within the limits

Using Z-table or technology for Normal Distribution, find the proportion of values that fall within the range Z = -1 to Z = 1. This reflects corks that are not defective. The value for this range is 0.6827.
04

Calculate the proportion of defective corks

Subtract the proportion found in Step 3 from 1 to get the proportion of corks that fall outside of the non-defective range (i.e., defective corks). This gives: 1 - 0.6827 = 0.3173. So, approximately 31.73% corks produced by this machine are defective.

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