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The z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. If the z-score is 0, it indicates that the data point's score is identical to the mean score. A z-score of 1.0 would indicate a value that is one standard deviation from the mean. Z-scores may be positive or negative, with a positive value indicating the score is above the mean and a negative score indicating it is below the mean.

For example, in the exercise given, finding the endpoint involved calculating the z-score based on the given area to the left (for part a) or right (for part b) of the endpoint. This approach to finding the z-score allows us to later convert it to actual values on the specific normal distribution curve by using the mean and standard deviation of the distribution.

The standard normal distribution is a special case of the normal distribution. It is used to compare scores from different normal distributions. A standard normal distribution has a mean of 0 and a standard deviation of 1. This special normal curve is referred to as the Z-distribution. Any normal distribution can be converted to a standard normal distribution using the z-score formula, effectively standardizing the data.

The computation of z-scores is based on this standardization process. Once the z-score is found using the standard normal distribution (as seen in the solution for the exercise), it can be used to determine where a score lies in relation to the overall distribution, often essential in statistical analysis and hypothesis testing.

The normal distribution, also known as the Gaussian distribution, is symmetric about the mean, and its shape corresponds to the well-known bell curve. There are several properties of the normal distribution curve that are essential to understand:

• The total area under the curve is 1, which represents the total probability of all possible outcomes.
• The mean, median, and mode of a normal distribution are all equal and located at the center of the distribution.
• Approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and nearly 99.7% falls within three standard deviations. This is known as the Empirical Rule or 68-95-99.7 rule.
• The tails of the normal curve extend infinitely in both directions without touching the horizontal axis, implying that there are always possibilities of extreme values (outliers).

When solving problems like the exercise provided, these properties help us understand the distribution of data and calculate the probabilities or areas under specific sections of the curve.