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Probability is the measure of how likely it is for a particular event to occur. In the context of a normal distribution, the probability is represented by the area under the curve for a specified range of values. For instance, if we want to determine the probability of a random variable falling between two values, we calculate the area under the distribution curve that lies between those values.

In the given exercise, we calculate the probability of the pH value being within certain intervals. Probability values range from 0 to 1, where 0 indicates impossibility and 1 indicates certainty. Therefore, when we say the probability that the pH value is between 5.90 and 6.15 is 0.7745, it means there is a 77.45% chance that a randomly selected pH value from the region would fall in that range.

A z-score, also known as a standard score, indicates how many standard deviations an element is from the mean. It's a way of standardizing scores on different scales so they can be compared directly.

In the exercise, we converted the pH values to z-scores to use the standard normal distribution table for finding probabilities. The formula to calculate a z-score is given by: \[ Z = \frac{X - \mu}{\sigma} \], where \(X\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. By comparing the z-scores, we are effectively comparing the relative position of values within the distribution.

The standard normal distribution is a specific instance of the normal distribution that has a mean of 0 and a standard deviation of 1. It is used to describe the distribution of z-scores. This distribution is symmetrical around the mean, and its shape is commonly known as the 'bell curve' due to its bell-like shape.

To find probabilities related to specific pH values in our exercise, we converted raw scores to z-scores and then used the standard normal distribution table. The areas under this distribution curve represent probabilities and are utilized to answer questions about the likelihood of random variables falling within certain ranges.

A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. There are two types of random variables: discrete and continuous. The pH in our exercise is an example of a continuous random variable, as it can take on any value within an interval.

Continuous random variables are represented by probability density functions (PDFs), like the normal distribution in this exercise. To calculate probabilities for intervals of continuous random variables, we find the area under the PDF curve corresponding to the interval. This concept is fundamental to understanding the probabilities associated with pH values in the soil sample problem.