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Z-scores are an essential part of understanding the standard normal distribution. They represent how many standard deviations an element is from the mean of the distribution. In a standard normal distribution, which is a special type of normal distribution with a mean of 0 and a standard deviation of 1, any data point can be converted to a z-score. This is important because it allows for the standardization of results, aiding in the comparison across different datasets.

For our exercise, the z-scores given are -2.18 and -0.42. These scores are less than the mean since they are negative. A negative z-score indicates that the data point is below the mean of the dataset. The specific values tell us precisely how far below the mean they are. For instance, a z-score of -2.18 means that the value is 2.18 standard deviations below the mean, while a z-score of -0.42 is 0.42 standard deviations below the mean.

Calculating the probability in a standard normal distribution involves finding the area under the curve between two z-score values. This is typically done using a Z-table, which lists z-score values alongside their corresponding probabilities. The probabilities represent the cumulative probability from the left end of the distribution up to a given z-score.

In our example, we need to find the probability that the random variable falls between -2.18 and -0.42. This requires us to look up each z-score in the Z-table. For -2.18, the probability is approximately 0.0146, and for -0.42, it is approximately 0.3372. To find the probability of falling between these two z-scores, we subtract the smaller probability from the larger one:

• Find the cumulative probability for each z-score.
• Subtract the lower cumulative probability from the higher one.
• Thus, the probability for our range is 0.3372 - 0.0146 = 0.3226.

This calculation tells us that there is a 32.26% chance of the variable falling within this range.

Shading the area under the normal curve helps in visually understanding probability. The standard normal curve is symmetric around the mean, which is 0. To represent probabilities on this curve, we often shade regions between specific z-score values, illustrating the concept of probability as an area under the curve.

In the given problem, the task is to shade the area between z = -2.18 and z = -0.42. This shaded area visually represents the probability we calculated, which is 0.3226. When shading this on a graph:

• Start at z = -2.18 and shade continuously to z = -0.42.
• The shaded area gives a visual impression of the probability and helps in reinforcing the concept of distribution spread.
• Understanding the shaded area offers insight into how likely a random variable will fall within that range under the standard normal distribution.

By matching the shaded area with the numerical probability, students can bridge the gap between mathematical theory and visual representation, strengthening their grasp of the concept.