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A z-score tells us how many standard deviations a data point is from the mean of a distribution. The mean of a standard normal distribution is 0, and its standard deviation is 1. If a data point has a z-score of -3.49, it means it is 3.49 standard deviations below the mean. Conversely, a z-score of 2.23 indicates the data point is 2.23 standard deviations above the mean.
To calculate a z-score, use the formula: \[ z = \frac{(X - μ)}{σ} \] where \(X\) is the data point, \(μ\) is the mean, and \(σ\) is the standard deviation. Z-scores help compare data points from different normal distributions. The standard normal table, or z-table, lists the cumulative probabilities for each z-score, which are essential for finding areas under the curve.

The area under the standard normal curve represents probabilities. Because the curve is symmetric around the mean (0), the total area under the curve sums to 1. The areas to the left or right of a z-score give probabilities of an event occurring less than or greater than that z-score. For example, to find the area to the right of \(z = -3.49\), look up the cumulative probability for \(z = -3.49\) (0.0002) and subtract it from 1: \(1 - 0.0002 = 0.9998\).
Similarly, the area to the right of \(z = -0.55\) is calculated by finding the cumulative probability for \(z = -0.55\) (0.2912) and subtracting from 1: \(1 - 0.2912 = 0.7088\). These calculations help in understanding how likely an event is to occur in a standard normal distribution.

Probability in the context of a standard normal distribution tells us how often a particular outcome is expected relative to the entire set of possible outcomes. In this setup:

• A z-score translates a specific data point to a location on the standard normal curve
• The area under the curve gives the cumulative probability
• By using z-scores and the standard normal table, you can find the probability of a random variable falling within a certain range.

For instance, to find the likelihood of a value being greater than \(z = 2.23\), you refer to the cumulative probability: \[1 - P(Z < 2.23) = 1 - 0.9871 = 0.0129\] This indicates that there's a 1.29% chance of getting a value greater than 2.23 standard deviations above the mean.
Similar steps can be done for \(z = 3.45\), showing that only about 0.03% of data points lie to the right of \(3.45\). This helps in probabilistic decision making and inferential statistics.