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A random variable is a fundamental concept in statistics that represents a numerical outcome of a random phenomenon. It is essentially a way to map outcomes to numbers. Random variables can be classified as either discrete or continuous.

• Discrete random variables: These have specific and countable outcomes, like the roll of a die (1, 2, 3, 4, 5, 6).
• Continuous random variables: These can take on any value within a given range, such as the measurement of time, temperature, or, in our case, the standard normal distribution.

In the standard normal distribution, the random variable is often denoted by the letter \( z \).
In this specific exercise, \( z \) is a random variable that follows a standard normal distribution, which has specific properties such as a mean of 0 and a standard deviation of 1. Understanding this concept helps us to predict the probability of different outcomes by using statistical tools like the Z-table.

Probability is the measure of the likelihood that an event will occur. It ranges from 0 (an impossible event) to 1 (a certain event). When dealing with continuous random variables, probability is often visualized as the area under a curve over a certain interval.
The standard normal distribution curve is bell-shaped and symmetric around the mean. Here, the probability corresponds to the area under the curve between specified values of \( z \). In the exercise, we needed to find the probability that \( z \) lies between -2.18 and -0.42.

• This is done by calculating the area under the curve between these two \( z \) values.
• The probability of \( z \) being less than a specific value was found using cumulative probabilities from a Z-table.
• The total probability for the desired range is calculated by finding the difference between two cumulative probabilities.

This calculated result represents the likelihood that the random variable \( z \) falls within this specific interval, hence solving the problem effectively.

Z-table, or standard normal table, is a crucial tool for finding probabilities in a standard normal distribution. It lists the cumulative probability of a standard normal random variable \( z \) being less than or equal to a given value. Here is how it works:

• The Z-table provides cumulative probabilities, meaning it shows the probability that \( z \) is less than or equal to a particular value.
• To find \( P(-2.18 \leq z \leq -0.42) \), first look up \( P(z \leq -0.42) \) and \( P(z \leq -2.18) \) in the Z-table.
• For our exercise, \( P(z \leq -0.42) \approx 0.3372 \) and \( P(z \leq -2.18) \approx 0.0146 \).
• The desired probability \( P(-2.18 \leq z \leq -0.42) \) is the difference between these cumulative probabilities: \( 0.3372 - 0.0146 = 0.3226 \).

The Z-table simplifies the calculation of probabilities for standard normal distributions, making it an indispensable tool in statistics. Understanding how to read and apply this table is essential for solving probability problems involving standard normal distributions.