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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.

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—thus, calculating the defective rate.

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 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.