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Probability calculation is a fundamental concept in statistics. It involves determining the likelihood of an event occurring. In the case of a standard normal distribution, the probability can be calculated as the area under the distribution curve.
To calculate probabilities for a range of values, like in our example, we often use the properties of the normal distribution curve. This curve is symmetric and bell-shaped, which means most values cluster around the mean, decreasing as you move away.
Using mathematical properties or software tools, we convert specific areas under this curve into probabilities. The area under the curve for values between two z-scores tells us the probability of a random variable falling within that range.

The z-score is a way of standardizing a data point with respect to the mean and standard deviation of a distribution. It's like a way of saying "how many standard deviations is this value away from the mean?"
For a standard normal distribution, the computation of a z-score is straightforward. Given that these distributions have a mean of 0 and a standard deviation of 1, the data point itself is the z-score. This makes calculations easier and faster.

• Positive z-score: The data point is above the mean.
• Negative z-score: The data point is below the mean.

The use of z-scores simplifies the probability calculations because they standardize different values, making it easier to compare results from different distributions.

Cumulative probability refers to the probability that a random variable is less than or equal to a certain value. In terms of the z-score, it's the probability that a variable falls to the left of a certain z-score on the standard normal distribution curve.
By looking at cumulative probability, you can calculate the likelihood of a range of outcomes. For intervals between two points, such as \(-1.12 < z < 1.75\), the cumulative probability at the higher z-score is subtracted by the cumulative probability at the lower z-score. This method is efficient for pinpointing the probability associated with any range within the distribution.
Utilizing a standard normal distribution table or statistical software can make finding cumulative probabilities at specific z-scores quick and accurate.