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When dealing with a normal distribution, Z-scores are a powerful tool for understanding how individual data points compare to the mean of the data set. A Z-score tells us how many standard deviations away a particular value is from the mean.

In simple terms, if you have a Z-score of 1, your value is one standard deviation above the mean. Conversely, a Z-score of -1 means the value is one standard deviation below the mean. The Z-score is calculated using the following formula:

• \( Z = \frac{X - \mu}{\sigma} \)

Here, \(X\) is the value you're examining, \(\mu\) is the mean of the distribution, and \(\sigma\) is the standard deviation.

In the context of our cork exercise, we used the Z-score formula to transform cork diameters of 2.9 cm and 3.1 cm into Z-scores. This standardization allows us to determine how many standard deviations these diameters are from the average size cork the machine is supposed to produce.

A standard deviation is a measure of how much the values in a data set vary or spread out from the mean. A smaller standard deviation indicates that the data points are close to the mean, while a larger standard deviation shows that the points are spread out over a wider range of values.

Calculating the standard deviation involves several steps, including finding the mean (average) of the data, subtracting the mean from each data point, squaring the result, finding the average of those squared differences, and finally taking the square root. In formulas, it's expressed as:

• \( \sigma = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} (X_i - \mu)^2 } \)

In our exercise, the standard deviation of 0.1 cm tells us that most of the cork diameters are close to the mean diameter, which is 3 cm. It gives insight into the typical deviation Cork diameters are expected to have under normal manufacturing conditions. Understanding the standard deviation helps in grasping how tightly or loosely the cork diameters are bound to their average size, influencing decisions about quality control and defect rates.

Probability is all about measuring the likelihood of an event occurring. In the world of normal distribution, it is often represented as the area under a curve in a graph.

The probability in a standard normal distribution is obtained using Z-scores. For our cork exercise, once we found the Z-scores for the 2.9 cm and 3.1 cm corks, we turned to a standard normal table, also known as the Z-table, to find corresponding probabilities.

• The probability that a cork is smaller than 2.9 cm corresponds to the area under the curve to the left of the Z-score for 2.9 cm.
• Likewise, the probability that a cork is larger than 3.1 cm is associated with the area to the right of the Z-score for 3.1 cm.

By adding these two probabilities, we could ascertain the overall proportion of corks that fall outside the specified range of 2.9 cm to 3.1 cm. This is useful because it quantifies not only the defective rate but potential costs involved in discarding or reprocessing non-compliant corks, ensuring product standards and customer satisfaction.