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Chapter 7: Problem 74

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 this machine that are defective is approximately 31.73%.

Step by step solution

01

Calculate the Z-scores

The z-score represents how many standard deviations an element is from the mean. It can be calculated using the formula: \( z = (X - \mu) / \sigma \), where \( \mu \) is the mean and \( \sigma \) is the standard deviation. For this exercise two z-scores will be calculated, one for the lower specification limit (2.9 cm), and one for the upper specification limit (3.1 cm). Let \( z_{2.9} = (2.9 - 3) / 0.1 = -1 \) and \( z_{3.1} = (3.1 - 3) / 0.1 = 1 \). This means that all the corks with diameters between 2.9 cm to 3.1 cm are within one standard deviation on either side of the mean.
02

Calculate the Proportion of Good Corks

Once the z-scores are calculated, the next step is to find the proportion of corks that lie within the given specifications. The z-table or standard normal distribution table is used to find the probability that a randomly selected cork has diameter between 2.9 cm and 3.1 cm. According to the Z-table, the proportion of data within one standard deviation (-1 to 1) of the mean is approximately 0.6827 or 68.27%.
03

Calculate the Proportion of Defective Corks

To get the proportion of defective corks: a cork can either be good or defective, so the probabilities of these two must add up to 1. So, the proportion of defective corks is given by \( 1-0.6827 = 0.3173 \) or 31.73%

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