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A Z-score is an essential concept in statistics often used to standardize data points relative to the mean of a dataset. Essentially, it tells us how many standard deviations a particular value is from the mean. This standardization allows us to compare scores from different datasets or different ranges.To calculate a Z-score, use the formula:

• \( z = \dfrac{X - \mu}{\sigma} \)

Where:

• \( X \) is the value of the element.
• \( \mu \) is the mean of the population.
• \( \sigma \) is the standard deviation of the population.

Calculating the Z-score translates any random value into a universal standard normal distribution, with \( \mu = 0 \) and \( \sigma = 1 \). This allows us to utilize standard normal distribution tables to find probabilities. In our original exercise, by finding Z-scores for values 20 and 30, we contextualized these specific scores within the broader distribution.

Standard deviation is a vital measure in statistics that quantifies the amount of variation or dispersion in a set of data values. A low standard deviation indicates that data points tend to be close to the mean, while a high standard deviation suggests more spread out data points.It can be mathematically represented as:

• \( \sigma = \sqrt{ \dfrac{1}{N} \sum_{i=1}^{N} (X_i - \mu)^2 } \)

Where:

• \( \sigma \) is the standard deviation.
• \( X_i \) are the individual data points.
• \( \mu \) is the mean of the data set.
• \( N \) is the number of data points.

In the context of our exercise, the standard deviation tells us about the spread of the distribution of our population. With \( \sigma = 5 \), we understand the extent to which the data values are expected to deviate from the mean. Understanding standard deviation is crucial, especially when standardizing data points to compute Z-scores.

Probability calculation in the context of a normal distribution helps determine how likely an event is to occur within a specified range. Once we have the Z-scores, these figures are used to determine the probabilities using a standard normal distribution table or calculator. To find the probability that a value will fall between two points, you can follow these steps:

• Calculate the Z-scores for the boundary values as shown in previous sections.
• Use a standard normal distribution table to find the probabilities associated with each Z-score.
• Subtract the lower probability from the higher probability to find the probability that a value falls between the two points.

In the original step-by-step solution, once the Z-scores for 20 and 30 were calculated as -1 and 1, corresponding probabilities (0.1587 and 0.8413) were found in the standard normal distribution table. By calculating the difference between these probabilities, we determined the likelihood of a value falling between them. The process of probability calculation becomes intuitive with a firm grasp of Z-scores and standard deviation.