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The standard normal variable, often represented as the Z-score, is crucial in converting any normal distribution into a standard normal distribution. This helps us simplify calculations by allowing us to use a universal reference, the standard normal distribution table.
The concept of the standard normal variable centers on the idea of standardizing a normally distributed random variable. A normal distribution can vary significantly with different means (\( \mu \)) and standard deviations (\( \sigma \)). By turning it into a standard normal variable, we set the mean to 0 and the standard deviation to 1. This is where the Z-score comes in, because it represents how many standard deviations an element is from the mean.

• A normal distribution is standardized using the formula: \[ Z = \frac{X - \mu}{\sigma} \]
• \( X \) stands for the random variable of interest.
• \( \mu \) represents the mean of the distribution.
• \( \sigma \) is the standard deviation.

Using the Z-score turns any normal distribution into the standard normal form, making it easier to compare values across diverse datasets by employing a common scale.

The Z-score is a statistical measurement that describes a value's relation to the mean of a group of values. It is measured in terms of standard deviations from the mean.
When you calculate a Z-score, you're essentially converting a raw score into a form that you can easily analyze using standard normal distribution tables.
The formula for calculating the Z-score is:
\[ Z = \frac{X - \mu}{\sigma} \]
Here, Z represents the Z-score, \( X \) is the raw score you are investigating, \( \mu \) is the mean of the population, and \( \sigma \) is the standard deviation.

• A positive Z-score means the data point is above the mean.
• A negative Z-score indicates it is below the mean.

By transforming raw data into a Z-score, you gain a deeper understanding of how the data point compares within its dataset and make calculations more manageable.

For our problem:

• Finding the Z-scores for 7 and 9 gives us a sense of where these values lie within the entire distribution.
• This enables the use of the next tool in our kit: the standard normal distribution table.

The standard normal distribution table, or Z-table, is a critical resource in statistics for finding probabilities associated with standard normal variables. This table compiles the cumulative probabilities of a standard normal distribution (mean = 0, standard deviation = 1) for various Z-score values.
The Z-table can tell you the probability of a value being less than a certain Z-score. For instance, a Z-score of 1.67 corresponds to a cumulative probability of approximately 0.9525 according to the Z-table, meaning that there's a 95.25% chance a randomly selected value from the distribution will be lower than 1.67.

• Locate the Z-score on the left-hand column and top row of the table.
• The intersection of these row and column entries gives the cumulative probability.

In practical applications:

• If you were looking at Z-scores of 1.67 and 3.33, you use the Z-table to find probabilities 0.9525 and 0.9996, respectively.
• The probability of a variable being between these two Z-scores is the difference in probabilities: 0.9996 - 0.9525 = 0.0471.

Using the standard normal distribution table streamlines the process of finding probabilities associated with any standard normal distribution, making it an indispensable tool in statistics.