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A Z-score is a statistical measurement that describes a value's position relative to the mean of a group of values. In simple terms, it tells us how far a particular value is from the average or mean.
To determine the Z-score, we use the formula:

• \( Z = \frac{x - \mu}{\sigma} \)

where:

• \( x \) is the value you have,
• \( \mu \) is the mean of the data, and
• \( \sigma \) is the standard deviation.

So if you have a Z-score of 1, it means that the value is one standard deviation above the mean. A negative Z-score, like -2.5, indicates that the value is below the mean.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It's a crucial concept in statistics because it tells us how much the data points differ from the average value.
Think of it like this: if the standard deviation is small, it means the data points are very close to the mean. If it’s large, the data points are spread out over a wider range.
In our example, the standard deviation is 0.06 inches. This tells us that the lengths of the rods vary slightly around the mean of 50 inches.
The formula to calculate standard deviation for a sample is:

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

where:

• \( \sigma \) is the standard deviation,
• \( N \) is the number of data points,
• \( x_i \) are the data points, and
• \( \mu \) is the mean.

A probability distribution is like a map that tells us how likely different outcomes are. It’s a mathematical function that provides the probabilities of occurrence of different possible outcomes.
One of the most common types of probability distributions is the normal distribution. It looks like a bell-shaped curve and describes how the values of a variable are distributed - most occur around the mean, and probabilities for values further from the mean taper off symmetrically.
For our machine example, the lengths of rods made exhibit a normal distribution with the mean (center) at 50 inches. This normal distribution means most rods are around 50 inches, but not exactly, varying slightly by the standard deviation.

Cumulative probability refers to the probability that a random variable will take a value less than or equal to a specific value. It's the sum of probabilities for all possible outcomes up to a certain point.
In our context, cumulative probability gives us the percentage of rods within a certain length range.
If we know the cumulative probability for a Z-score from a table, we can determine the percentage of rods falling within a given length using:

• For Z-scores of -2.5 and 2.5, the cumulative probabilities are found as 0.0062 and 0.9938 respectively.
• The difference gives us the probability of lengths being between these values, which is 98.76%.

Thus, subtracting from 1 gives the percentage of rods beyond these lengths. In this exercise, that's 1.24% of rods being discarded.