91Ó°ÊÓ

The mean, often denoted by the symbol \( \mu \), is the average or central value of a set of data in probability and statistics. In the context of a standard normal distribution, the mean is always 0. This zero-mean feature of the standard normal distribution allows for simplified computations and is a cornerstone when using the normal distribution for statistical analyses.

Think of the mean as a balance point. If you were to spread out all the values from a data set along a number line, the mean would be the point where this line is perfectly balanced. For a standard normal distribution, everything revolves around zero. This is crucial because it means that 50% of the data lies below the mean and 50% lies above. It helps us in calculating probabilities using the z-table as we'll see in later sections.

The standard deviation, denoted by \( \sigma \), measures the amount of variation or dispersion in a set of values. Specifically, it tells you how spread out the data points are from the mean. In a standard normal distribution, the standard deviation is not just any number - it is precisely 1.

A standard deviation of 1 in a standard normal distribution indicates that data points typically fall within one unit of distance from the mean, or zero. This gives a nice, clean measure of spread which can be used intuitively. When considering areas within a standard deviation of the mean, you are essentially measuring how much of your data is close to the average value. In our example case, it asks for the area between -1 and 1, which represents one standard deviation away from the mean in both directions.

The z-table, or standard normal distribution table, is a mathematical table that allows you to find the cumulative probability of a standard normal distribution below a particular z-score. It serves as a reference to locate areas under the standard normal curve. The value in a z-table corresponds to the area under the standard normal probability curve to the left of any given z-score.

For example, when you look up a z-score of 1 in the z-table, you find a value of about 0.8413. This value indicates that approximately 84.13% of the data in a standard normal distribution lies below a z-score of 1. Without the z-table, calculations of probabilities involving the normal distribution would be much more complex. It is particularly useful in our problem to determine the area between two z-scores by finding the cumulative areas for each and subtracting them.

Cumulative area under the normal distribution curve is the probability that a randomly selected value from the distribution is less than or equal to a particular value. In terms of a standard normal distribution, it tells us how much area, as a proportion of the total, lies to the left of a given z-score.

To find the area between two z-scores, such as in our problem, you need to calculate the cumulative area up to the rightmost z-score and subtract from it the cumulative area up to the leftmost z-score. In our example, the cumulative area to the left of z = 1 is approximately 0.8413, and to the left of z = -1 is approximately 0.1587. Subtracting these values, as shown in the step-by-step solution, gives us the total probability or cumulative area between these two scores, which is approximately 68.26%.