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Understanding the Z-score is key to navigating problems involving the Standard Normal Distribution. The Z-score converts different data points into a standard form, showing how far a point is from the mean in terms of standard deviations.

• To calculate a Z-score, use the formula: \( z = \frac{(X - \mu)}{\sigma} \), where \( X \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
• A Z-score of 1 indicates a value 1 standard deviation above the mean. Conversely, a Z-score of -1 indicates a value 1 standard deviation below the mean.

Z-scores help us standardize different datasets, making them comparable on a broader level. In our exercise, understanding Z-scores allows us to use standard normal distribution tables to find probabilities associated with specific data points.

The standard deviation is a measure that tells us how spread out the data is around the mean. It's a crucial concept in statistics, particularly in the context of normal distributions.

• A smaller standard deviation means that data points tend to be closer to the mean.
• A larger standard deviation indicates that the data is more spread out.

In a standard normal distribution, the mean is 0, and the standard deviation is 1. This gives us a benchmark to understand how far or close observations are in terms of standard deviations. By knowing the standard deviation, one can quickly determine where an observation lies within a data set by using the Z-score.

A Probability Table, specifically the Z-table or Standard Normal Distribution Table, helps us find the area under the curve of a normal distribution. This area represents probability and is essential in determining the likelihood of a given observation.

• The Z-table lists cumulative probabilities, which are the probabilities that a random variable is less than a particular Z-score.
• To find an area to the right of a Z-score, subtract the cumulative probability from 1.

For instance, in the exercise, finding the probability of a variable being at least 1 standard deviation above the mean involves looking up the Z-score of 1. The table gives a cumulative probability of 0.8413. Therefore, the probability of being at least 1 standard deviation above is \( 1 - 0.8413 = 0.1587 \). This process is repeated for other Z-scores to find probabilities of different observations.

Tail Probability refers to the probability of a variable falling into the "tails" of a probability distribution, which are the extreme ends on either side. In the context of a standard normal distribution, we're interested in how often data points land more than a certain number of standard deviations away from the mean.

• The right tail probability is found by looking beyond a positive Z-score.
• The left tail probability looks beyond a negative Z-score.

In the exercise, to determine the probability of an observation being at least 1 standard deviation from the mean, whether above or below, tail probabilities are calculated. Using the symmetry of the normal distribution, both tail probabilities for Z-scores of 1 and -1 are the same, each yielding a probability of 0.1587. Sketching a curve with shaded tails offers a powerful visual to solidify understanding.