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When working with a set of data, one of the ways to tell how an individual data point compares to the average is by using a z-score. The z-score is a numerical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. If a z-score is 0, it is at the mean. A positive z-score indicates the value is above the mean and a negative score says it's below. For example, if the area to the left of a z-score is approximately 0.6666, we're lookng at a point that's higher than 66.66% of all data points in the normal distribution.

To calculate a z-score you'd use the formula:
\[\[\begin{align*}\( z = \frac{{X - \mu}}{{\sigma}} \)\end{align*}\]\]
Here, \(X\) is the value you are interested in, \(\mu\) is the mean of the dataset, and \(\sigma\) is the standard deviation. This calculation will yield a score which helps in understanding where the value stands in comparison to the rest of the data set.

The Normal curve, also known as the Gaussian curve or bell curve, represents the standard normal distribution graphically. It's symmetric around its mean and has a single peak right at the center, where the mean, median, and mode of the distribution are all equal. The spread of a Normal curve is determined by its standard deviation, which dictates how dispersed the values are around the mean.

Standard normal distribution specifically has a mean of 0 and a standard deviation of 1. The percentage of data within certain ranges can be determined through the properties of the Normal curve. For example, about 68% of the data falls within one standard deviation, 95% within two standard deviations, and 99.7% within three standard deviations of the mean.

In the context of the provided exercise, the Normal curve helps visualize how much of the data falls below the z-score of approximately 0.43.

The concept of cumulative probability is tied to the Normal curve and z-scores. It refers to the probability that a variable will take a value less than or equal to a certain number. In the standard normal distribution, it is the area under the curve to the left of a specific z-score.

For instance, a cumulative probability of 0.6666 means that there is a 66.66% chance that a randomly selected score from the distribution is less than or equal to the associated z-score. Cumulative probability is fundamental when addressing the question of 'how much of all possible outcomes fall below a specific threshold?' and is practically depicted as the area under the normal curve to the left of a z-score point.

Sometimes, you know the cumulative probability (the area under the Normal curve to the left of the z-score) but need to find the corresponding z-score. This process is known as a reverse lookup in z-table. A z-table, or standard normal table, shows the cumulative probability for z-scores along the Normal curve.

To perform a reverse lookup, you start with the cumulative probability and search the table for the closest value, then locate the corresponding z-score. This z-score is the number of standard deviations away from the mean. In the exercise at hand, to find a z-score that corresponds to a cumulative probability of 0.6666, you would look through the z-table, locate the area closest to that value, and then determine the z-score that aligns with it. Adding up the values as the exercise suggests (0.4 and 0.03), you arrive at the z-score (0.43) connected to the 66.66% cumulative probability.