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The Z-table, or standard normal distribution table, is an essential tool in statistics used to find probabilities associated with the standard normal distribution. The standard normal distribution is a bell-shaped curve that is symmetric around the mean, which is zero.

The Z-table helps you determine the area under this curve to the left of any given z-value. Since the total area under the curve is equal to 1, it represents 100% probability. The Z-table typically shows cumulative probabilities from the mean to a specific z-score.

When you refer to the Z-table:

• Look at the row and column that correspond to your z-score. For example, for a z-score of -0.96, find the row labeled -0.9 and the column labeled 0.06.
• The number found at this intersection gives the probability to the left of the z-score, meaning it's the cumulative probability.

Understanding how to read the Z-table is crucial for finding probabilities of events within a normal distribution.

Cumulative probability is a concept that describes the probability that a random variable is less than or equal to a particular value. For a standard normal distribution, it indicates the area under the curve to the left of a specified z-score.

To find these cumulative probabilities, we use the Z-table:

• For any z-value, the cumulative probability shows how much area lies to the left of that value on the standard normal distribution curve.
• For instance, a cumulative probability of 0.5 means that half of the distribution's area falls below this point, often corresponding to a z-score of 0.

In our example, the cumulative probability for z = -0.96 is approximately 0.1685, showing that there's a 16.85% chance of falling below this z-score. Similarly, for z = -0.36, the cumulative probability is around 0.3594, meaning 35.94% of the distribution is below this point.

Being adept with cumulative probability is key, as it helps in assessing and calculating probabilities over specific intervals in a normal distribution.

Probability calculation in the context of the standard normal distribution involves determining the probability that a random variable falls within a specific range of z-scores. Using cumulative probabilities, you can compute these interval probabilities effectively.

To find the probability within an interval between two z-scores, such as between z = -0.96 and z = -0.36:

• Identify the cumulative probabilities for each z-score using the Z-table.
• Subtract the smaller cumulative probability from the larger one to get the probability of the variable being between these z-scores.

For our problem, with cumulative probabilities 0.1685 for z = -0.96 and 0.3594 for z = -0.36, the calculation is:\[0.3594 - 0.1685 = 0.1909\]This result tells us that there's a 19.09% chance the random variable lies between z = -0.96 and z = -0.36.

Mastering this calculation process allows you to analyze various intervals within the standard normal distribution effectively, a fundamental skill in statistics.