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The Z-score, also known as the standard score, is a critical concept in statistics when dealing with normal distributions. It helps us determine how far away a particular data point is from the mean in terms of standard deviations.
To calculate the Z-score for a given data point, we use the formula:

• \[ Z = \frac{X - \mu}{\sigma} \]

• \(X\) is the data point in question,
• \(\mu\) is the mean of the distribution,
• and \(\sigma\) is the standard deviation of the distribution.

The Z-score tells us how many standard deviations a data point (\(X\)) is from the mean. A positive Z-score indicates that the data point is above the mean, while a negative Z-score shows it's below the mean. For example, a Z-score of -4, as calculated for the value 140 from the exercise, indicates that 140 is 4 standard deviations below the mean of 160.

The standard normal distribution is a special case of the normal distribution. It has a mean of 0 and a standard deviation of 1. This distribution is represented by the Z-scores. The standard normal distribution allows us to standardize different normal distributions and apply to them a single set of probabilities.
When we transform a normal distribution into a standard normal distribution using the Z-score formula, we're effectively converting the data points into Z-scores.
These Z-scores then tell us where the data points lie in terms of standard deviations relative to the mean.
The standard normal distribution table, or Z-table, is a tool that helps find the probability of a Z-score being less than (to the left of) a given value. For instance, a Z-score of 8 would be far to the right on the standard normal distribution curve, meaning it covers almost all typical values so its probability to the left is practically 1.

Calculating areas under the normal distribution curve is key when trying to find out the probability of certain events. These areas represent probabilities.
For a distribution \(N(\mu, \sigma^2)\), areas under the curve can tell us about the proportion of data points that lie above or below a certain value.
For example, in the exercise:

• The area to the right of a Z-score represents the probability of a value being greater than the Z-score.
• The area to the left indicates the probability of a value being less or equal to the Z-score.

We often use tables of the standard normal distribution to find these areas. For instance, for a Z-score of -4, the area to the left is very small (0.00003), and the area to the right can be calculated as 1 minus this value (0.99997).
This demonstrates how extremely rare it is to find such a low score compared to the mean in a normal distribution.