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Calculating a z-score is the first step when dealing with problems involving normal distribution. A z-score tells you how many standard deviations a particular data point is from the mean. This can be useful in determining how unusual or typical a data point is within a given distribution. To calculate a z-score, use the formula: \( z = \frac{X - \mu}{\sigma} \)

• \(X\) refers to the data point you are examining.
• \(\mu\) is the mean of the distribution.
• \(\sigma\) is the standard deviation.

For instance, if you have a score of 405 from a distribution with a mean of 380 and a standard deviation of 20, the z-score is calculated as: \( z = \frac{405 - 380}{20} = 1.25 \). This z-score of 1.25 indicates that 405 is 1.25 standard deviations above the mean of this distribution.

A probability distribution describes how probabilities are distributed over the possible values of a random variable. Normal distribution is a specific kind of probability distribution that is characterized by its bell-shaped curve.

In a probability distribution:

• The mean is the measure of central tendency, giving the center of the distribution.
• The standard deviation measures the dispersion or spread of the distribution.

For a normal random variable like our example with \( \mu = 380 \) and \( \sigma = 20 \), the distribution is symmetric around the mean. Most data points lie around the mean, and as you move further from the mean, the probabilities decrease. This property is crucial for calculating probabilities, which we often do by converting data points to z-scores and using standard normal distribution tables.

The standard normal distribution is a special type of normal distribution with a mean of 0 and a standard deviation of 1. It's often used as a reference point for measuring other normal distributions.

Why do we use this? Because the standard normal distribution has a consistent z-table, a tool used to find probabilities associated with z-scores. Once you have a z-score, you can consult the z-table to determine the probability of the data point occurring within the standard normal distribution.

For example, if the z-score of a data point is 1.25, you'd look it up in the z-table to find the probability \( P(Z < 1.25) \), which can give important insights into where a data point lies in relation to the entire distribution. This makes the standard normal distribution a convenient tool for analyzing data across various distributions.

Cumulative probability refers to the probability that a random variable falls within a particular range. For a normal distribution, it's the probability that a variable is less than or equal to a given value, denoted as \( P(X \leq x) \).

When working with z-scores, cumulative probability involves determining the area under the curve to the left of a specific z-score. This is important because the standard normal distribution table typically provides cumulative probabilities.

• For \( P(X < 405) \), \( P(Z < 1.25) \) gives a cumulative probability of approximately 0.8944.
• If you want the probability over a range like \( P(400 < X < 430) \), you calculate the difference between two cumulative probabilities: \( P(Z < 2.5) - P(Z < 1) \).
• For \( P(X > 400) \), use the complement rule: \( P(Z > 1) = 1 - P(Z < 1) \).

This calculation approach is powerful for interpreting data outcomes within real-world contexts.