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A z-score is a statistical measurement that describes a value's position relative to the mean of a group of values. It is a crucial concept in statistics, especially in the context of the standard normal distribution. When we calculate a z-score, it tells us how many standard deviations away from the mean a particular value is. To determine a z-score, you use the formula: \[ z = \frac{X - \mu}{\sigma} \]where:

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

A positive z-score indicates that the value is above the mean, while a negative z-score shows it is below the mean.
Understanding how z-scores work helps us transform different data points into a standard form, making it easier to compare them against a standard normal distribution.

The Z-table, also known as the standard normal table, is a mathematical table used to find the probability that a standard normal random variable falls to the left of or below a given z-score. It is a valuable tool because it allows us to convert z-scores into probabilities, showing the portion of the distribution that is accounted for by that score. When using a Z-table:

• You need to know the z-score for which you are calculating the probability.
• Navigate the Z-table to find the corresponding probability.

In our solution, the Z-table was used to find the area to the left of z = 1.23 and z = 1.90, which are 0.8907 and 0.9713, respectively.
Once the probabilities are located, it's possible to easily find the area between two z-scores by subtraction, which gives us the desired probability between these scores.

Probability in the context of the standard normal distribution refers to the likelihood that a random variable falls within a particular range of values. It's expressed as a number between 0 and 1, where 0 means the event will not occur, and 1 means it certainly will. In the process of finding the area under the curve between two z-scores, what we're essentially calculating is the probability that a standard normal random variable will fall within these bounds. Our final result, 0.0806 in this case, represents an 8.06% chance that a randomly selected variable from this distribution will lie between z = 1.23 and z = 1.90.
Understanding probability helps in interpreting a range of scenarios, such as predicting outcomes and making decisions based on statistical data.

The area under the curve in a standard normal distribution is of vital importance in probability and statistics. This area represents the cumulative probability of a distribution, which means it shows how likely a certain range of outcomes is. For instance:

• The total area under the curve equals 1, reflecting the fact that the total probability of all possible outcomes must equal 1, or 100%.
• The area to the left of a specific z-score gives the probability of a value being less than this score.

In the exercise, the area between z = 1.23 and z = 1.90 was found to be 0.0806. This indicates that 8.06% of the data falls within this range due to the bell shape of the normal distribution.
Understanding how to calculate and interpret these regions is crucial for making predictions and decisions based on data analysis.