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In statistics, a z-score helps us determine how far a specific data point is from the mean, measured in terms of standard deviations. The z-score can be positive or negative based on whether the data point is above or below the mean.
This concept becomes crucial when working with the standard normal distribution—a normal distribution with a mean of 0 and a standard deviation of 1.
In the original problem, we find z-scores by identifying areas of interest under the curve. These areas are connected to probabilities showing how often data points fall below a certain level.
By using a standard normal distribution table or a calculator, you can look up the z-score related to a specific area. For example, an area of 0.1965 under the curve corresponds to a z-score of about 0.5.
The z-score tells us that this point is 0.5 standard deviations above the mean of the distribution.

The normal curve, also known as the bell curve, is a visual representation of a normal distribution. It is symmetrical, with most data points clustering around the mean. As we move away from the mean, data points become less frequent, forming the characteristic bell shape.
The curve is defined by its mean and standard deviation, and when standard, the mean is zero with a standard deviation of one.

• Mean (\( ext{μ}\)): Central point of the curve, where it peaks.
• Standard Deviation (\( ext{σ}\)): Measures how spread out the data is.

When dealing with a standard normal curve, we're often interested in finding what percentage of data points lie beyond a certain z-score, which is where we use tables or calculators.

The left tail of the normal curve represents the area to the left of a specific z-score, often associated with values farther below the mean.
This area represents the probability that a randomly selected data point falls below the specified z-score.
In the original problem, when it specifies the left tail with an area of 0.2050, it refers to this probability. To find the z-score for this area, you look for (1 - 0.2050 = 0.7950) in a standard normal distribution table. The corresponding z-score is about -0.84, showing it is below the mean.
The left tail can be useful for identifying lower-bound probabilities or values significantly lower than the average.

The right tail of the normal curve corresponds to the area right of a specific z-score, representing higher values above the mean.
This region accounts for the probability that a data point is greater than the chosen z-score.
For example, in the problem statement, it refers to the right tail with an area of 0.1053. To find the associated z-score, we look at the complement of this area (1 - 0.1053 = 0.8947) and find it in a standard normal table. This results in a z-score of approximately 1.26.

• Values in the right tail are higher than the mean.
• Useful for assessing upper probability bounds or detecting unusually high values in dataset distribution.

Understanding the right and left tails helps determine where data points fall within a distribution and informs decision-making based on probability.