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The z-score is a statistical measure that tells us how many standard deviations a data point is from the mean. It is a way to standardize scores across different datasets or different metrics.

• To calculate a z-score, you need three pieces of information: the data point (\( x \)), the mean (\( \mu \)), and the standard deviation (\( \sigma \)).
• The formula is: \[ z = \frac{x - \mu}{\sigma} \]

A z-score can be positive or negative. A positive z-score means the data point is above the mean, while a negative z-score means it is below the mean. Here, a z-score of -2 implies that the data point is 2 standard deviations below the mean.
Understanding z-scores is important because they allow us to determine the probability of a data point falling within a particular range in a normal distribution. This is crucial when analyzing data to make informed decisions.

The standard normal distribution is a special normal distribution that has a mean of 0 and a standard deviation of 1. It is represented by the bell-shaped curve known as the Gaussian distribution.

• Every normal distribution can be transformed into a standard normal distribution using z-scores.
• When working with the standard normal distribution, probabilities are easier to find because we can use a standard normal distribution table that gives the likelihood of a z-score being less than a given value.

The transformation to a standard normal distribution allows you to compare different sets of data points by converting them to a common scale. Understanding the properties of the standard normal distribution is essential for calculating probabilities and conducting statistical analyses effectively.
If you're ever unsure about a specific probability, consulting the standard normal distribution table or using statistical software will help you find the probability related to your calculated z-scores.

Interval probability refers to the probability that a random variable falls within a specific range or interval. This is often expressed as \( P(a \leq X \leq b) \) where \( a \) and \( b \) are the endpoints of that range.

• Interval probabilities are evaluated by finding the probabilities associated with the z-scores of these endpoints.
• After converting the original data points into z-scores, the next step is to use the probabilities derived from the standard normal table to identify the likelihood of a variable being within the specified interval.

The key takeaway is to subtract the smaller cumulative probability from the larger one to find the interval probability. This process requires understanding both z-scores and the standard normal distribution.
In practical terms, this helps in determining the likelihood of occurrences within a specific interval, which is essential in fields like finance, quality control, and other areas utilizing statistical inference.