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A Z-table, also known as the standard normal distribution table, is a handy tool in statistics. It allows you to find the percentage of values below a particular z-score in a standard normal distribution.
A z-score tells us how many standard deviations away from the mean a particular data point is. The Z-table typically lists these scores in the leftmost column and their cumulative probabilities for each score in the corresponding row.
Using a Z-table is straightforward:

• Locate the z-score you need. In our exercise, we need the scores for 0 and -0.82.
• Find the corresponding cumulative probabilities. For z = 0, it is 0.5 (since 0 is the mean of the distribution). For z = -0.82, look up the table to find approximately 0.2061.

The Z-table is a powerful way to convert z-scores into probabilities, making it integral to solving standard normal distribution problems.

Cumulative probability is the probability that a random variable is less than or equal to a particular value. When we deal with standard normal distributions, we often want to know how much of the distribution lies below a certain z-score.
This is essential for interpreting data points in terms of their relative standing within the distribution. For a standard normal distribution with a mean of 0 and standard deviation of 1, cumulative probabilities help us understand the likelihood of observing a value below a certain point.

• For example, the cumulative probability of z = 0 is 0.5, indicating that half of the distribution lies below this point.
• If we find that the cumulative probability of z = -0.82 is 0.2061, it tells us that 20.61% of the distribution lies below this value.

Understanding cumulative probabilities is crucial because they give context to where data points stand on the normal distribution curve.

Calculating a probability range in a standard normal distribution involves determining the probability that a random variable falls within a specific set of bounds, such as between two z-scores.
To find this probability, we rely on the cumulative probabilities linked with these z-scores. Here's how you can calculate it:

• First, find the cumulative probabilities for the upper and lower z-scores using a Z-table.
• Subtract the cumulative probability of the lower z-score from the cumulative probability of the upper z-score.

For our problem: - The cumulative probability of z = 0 is 0.5.- The cumulative probability of z = -0.82 is approximately 0.2061.
Thus, the probability range for -0.82 ≤ z ≤ 0 is calculated as: \[ P(-0.82 \leq z \leq 0) = P(z \leq 0) - P(z \leq -0.82) = 0.5 - 0.2061 = 0.2939 \] This tells you that there is a 29.39% chance that a data point lies between -0.82 and 0 on a standard normal curve. Understanding this calculation helps in assessing how likely a data point resides in any given section of a distribution.