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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 diameter of the corks follows a normal distribution with a mean of 3 cm and a standard deviation of 0.1 cm. By calculating the Z-scores for the lower and upper bounds of the specifications (2.9 cm and 3.1 cm), we find the probabilities associated with these Z-scores and use them to determine the proportion of corks within the specified range (about 68.26%). Subtracting this proportion from 1, we find that about 31.74% of the corks produced by this machine are defective.

Step by step solution

01

Identify the parameters of the normal distribution.

The diameter of the corks has a normal distribution, with a mean (\(\mu\)) of 3 cm, and a standard deviation (\(\sigma\)) of 0.1 cm. So, the distribution can be represented as \( N(\mu, \sigma^2)=N(3, 0.1^2) \).
02

Calculate the Z-scores for the lower and upper bound of the specifications.

In order to find the proportion of defective corks, we need to find the Z-scores associated with diameters 2.9 cm and 3.1 cm. The Z-score can be calculated using the formula \(Z = \frac{x - \mu}{\sigma}\), where \(x\) represents the value for which we are calculating the Z-score. For the lower bound (2.9 cm): \(Z_{2.9} = \frac{2.9 - 3}{0.1} = -1\) For the upper bound (3.1 cm): \(Z_{3.1} = \frac{3.1 - 3}{0.1} = 1\)
03

Find probabilities associated with the Z-scores.

Next, we will use a standard Z-table or a calculator to find the probabilities associated with the calculated Z-scores. For \(Z_{2.9} = -1\), the probability \(P(Z < -1) = 0.1587\). For \(Z_{3.1} = 1\), the probability \(P(Z < 1) = 0.8413\).
04

Calculate the proportion of defective corks.

To find the proportion of defective corks (those with a diameter outside the specified range), we will subtract the total probability of corks within the range from 1. The probability of corks within the specified range is given by: \(P(2.9 \le \text{diameter} \le 3.1) = P(Z < 1) - P(Z < -1) = 0.8413 - 0.1587 = 0.6826\). The proportion of defective corks is then: \(1 - P(2.9 \le \text{diameter} \le 3.1) = 1 - 0.6826 = 0.3174\). So, about 31.74% of the corks produced by this machine are defective.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. It quantifies how much the numbers in a data set devary from the mean on average. A low standard deviation means that most of the numbers are close to the mean, while a high standard deviation indicates that the numbers are more spread out.

For the cork-cutting machine, a standard deviation of 0.1 cm means that the diameters of the corks tend to deviate from the mean (3 cm) by an average of 0.1 cm. This small standard deviation suggests that the machine's performance is fairly consistent, with corks' diameters clustering near the target size. However, even with a small standard deviation, there can still be a significant defective rate if the acceptable size range is tight.
Z-score
The Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations from the mean. It is calculated using the formula Z = (x - \( \mu \) ) / \( \sigma \) , where x is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

Z-scores are a critical part of understanding the probability of an event occurring within a normal distribution. In the case of the cork dimensions, Z-scores were calculated for the upper and lower specification limits to identify what percentage of corks falls outside the acceptable range鈥攖hus, calculating the defective rate.
Defective Rate Calculation
Calculating the defective rate involves identifying the proportion of output that fails to meet the established quality standards. In the context of the cork-cutting machine, the defective rate represents the percentage of corks that are not within the specified diameter range of 2.9 to 3.1 cm.

To find this, after calculating the Z-scores for the specification limits, we use the area under the normal distribution curve outside the range defined by these Z-scores. The area between the Z-scores corresponds to the proportion of corks that meet the specifications, and by subtracting this from 1, we obtain the defective rate. This approach effectively helps us quantify the machine's performance and its capability to produce corks that fulfill the quality standards.
Probability Distributions
Probability distributions describe how the values of a random variable are distributed. For continuos variables, the most known probability distribution is the normal distribution, also known as the Gaussian distribution, which is bell-shaped and symmetric about the mean.

When dealing with quality control, such as monitoring the diameter of corks, the normal distribution is often used because many processes naturally follow this pattern due to the Central Limit Theorem. The normal distribution is defined by two parameters: its mean (the center of the distribution) and its standard deviation (which determines the width of the distribution). The area under the curve between two points gives us the probability of a random variable falling within that range, which is fundamental for calculating proportions like the non-defective rate of corks.

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Most popular questions from this chapter

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