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The normal distribution is a continuous probability distribution that is symmetrical, meaning most observations cluster around the central peak, and the probabilities for values taper off symmetrically on both sides. It's also known as a bell curve due to its shape. Normal distributions are defined by two parameters:

• Mean (\( \mu \)): This is the average or expected value of the data points.
• Standard Deviation (\( \sigma \)): This measures the spread of the data points around the mean.

In the context of the GRE scores mentioned in the exercise, the verbal and quantitative scores are modeled as normal distributions with given means and standard deviations. This allows us to make predictions about where a student's scores are likely to fall within the distribution of all test-takers.

Independent events are those whose outcomes do not affect one another. In probability, two events are independent if the occurrence of one does not influence the probability of the other occurring.

For example, when rolling two dice, the result of one die does not affect the result of the other die. In our exercise, the verbal and quantitative sections of the GRE are independent. This means a student's performance on one section does not impact their performance on the other.

This independence allows for simplified calculations when combining the scores from these sections. For the combined score's variance, we add the variances of each section (since they are independent), thereby simplifying the calculations.

A z-score is a statistical measurement that describes a data point's relation to the mean of a group of values. It shows how many standard deviations a particular value is from the mean, capturing whether it is above or below the average.

The formula for calculating a z-score is:\[ z = \frac{X - \mu}{\sigma} \] where \( X \) is the value being assessed, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

• A positive z-score indicates the value is above the mean.
• A negative z-score indicates it is below the mean.

Using z-scores, you can determine probabilities and percentiles within the normal distribution. In the original problem, the z-score helps us find the probability of a combined GRE score exceeding a certain value and identifying percentile ranks.

A percentile is a measure used in statistics to indicate the value below which a given percentage of observations fall. For example, the 90th percentile is the value below which 90% of the observations can be found.

Percentiles are useful in comparing scores and are often used to understand where an individual's score stands relative to others.

To find the percentile of a particular score in a normal distribution, you first find the z-score and then use the standard normal distribution to determine the corresponding percentile.

• For example, a z-score of approximately 1.28 corresponds to the 90th percentile, meaning that if a student scores this high or higher, they performed better than 90% of test-takers.

Percentiles are hugely valuable in educational testing and other fields to evaluate comparative performance.