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A Z-score is a statistical measure that gives you an idea of how far a particular data point is from the mean, in terms of standard deviations. It helps in standardizing different data sets and comparing them using a common measure. By using the formula \( z = \frac{x - \mu}{\sigma} \), where

• \( x \) is the value of interest.
• \( \mu \) is the mean of the data set.
• \( \sigma \) is the standard deviation of the data set.

You convert any normal distribution into a standard normal distribution, which has a mean of 0 and a standard deviation of 1. This transformation simplifies the process of finding probabilities, as it allows us to use the standard normal (Z) table. For example, in the given problem, calculating Z-scores for 28 and 12, with mean 15 and standard deviation 5, yields scores of 2.6 and -0.6, respectively.

The utility of Z-scores becomes evident when determining probabilities for the tails of the distribution curve. A higher absolute Z-score indicates a value further from the mean, making it more unusual if the data are normally distributed.

Probability in a normal distribution relates to the area under the curve, representing how likely different outcomes are. Once you calculate a Z-score for a specific value, you use a Z-table (also known as a standard normal distribution table) to find the probability of observing less than or more than your specified value.

In the context of the exercise:

• For point (a) of the exercise, with a Z-score of 2.6, looking up this value in a Z-table provides a cumulative probability of 0.9953.
• For point (b), a Z-score of -0.6 correlates to a cumulative probability of 0.2743.

These values help determine the proportion of the dataset falling to the left or right of a given point. Importantly, probabilities allow conclusions based upon real-world expectations, and predictively model likely future outcomes.

When seeking the area to the right of a particular Z-score, you subtract the table probability from 1. Conversely, the table provides the probability directly if you're looking to the left.

A standard normal distribution is a special type of normal distribution that has been standardized: its mean is 0 and its standard deviation is 1. This is achieved through the transformation of raw scores (data points) to Z-scores.

This transformation is crucial because it allows for the use of the standard normal (Z) table, simplifying the process of determining probabilities. Each value in the table corresponds to a cumulative probability that a random variable with a standard normal distribution is less than or equal to a given Z-score.

• The standard normal distribution is symmetric, and its total area is equal to 1.
• The central point represents the mean, median, and mode, making it a powerful tool in statistics for evaluating data relative to a central point.

Understanding and applying the standard normal distribution allows statisticians and researchers to compare different data sets with ease. By converting any normal curve into the standard form, the complexity of calculations is reduced, and data analysis becomes more streamlined.