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The concept of z-scores is fundamental in understanding the standard normal distribution. A z-score tells you how many standard deviations a data point is from the mean of a distribution. In a standard normal distribution, the mean is 0 and the standard deviation is 1. Knowing this, a z-score of -1.46 means the data point is 1.46 standard deviations below the mean. Conversely, a z-score of 1.46 would be 1.46 standard deviations above it. Z-scores are particularly useful because they allow you to compare scores from different normal distributions. By converting scores to a universal z-score, you can determine where a score stands in relation to others in a standard normal model. When tackling exercises that involve finding the area under a normal curve, z-scores help specify the data points of interest.

Cumulative probability is the probability that a random variable will assume a value less than or equal to a given number. When dealing with the standard normal distribution, cumulative probability provides the total area under the curve to the left of a z-score. This is a critical step when calculating the probability between two points. For instance, if you have a z-score of -1.46, the cumulative probability is about 0.0721. This means there's a 7.21% chance a random variable will be less than or equal to this z-score. By comparing cumulative probabilities of different z-scores, you can find the probability between any two points. You simply subtract the cumulative probability of the smaller z-score from the larger one to find this area. This process is essential when determining probabilities for specific ranges within a normal distribution.

A standard normal table, also known as a z-table, is an essential tool for finding cumulative probabilities in a standard normal distribution. It provides the area under the curve from the extreme left of the curve up to any desired z-score. These tables are arranged to help you quickly locate the probability associated with specific z-scores. Looking up values is straightforward but requires careful attention to detail, as every decimal is significant.

• Find your desired z-score in the leftmost column of the table.
• Align it with the correct column head to read the cumulative probability.

Using the table, the cumulative probability for a z-score of -1.46 is 0.0721, while for -1.77, it is approximately 0.0384. These values from the standard normal table help you calculate how much of the distribution lies between any two points, making them indispensable for understanding probabilities in normal distributions.