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A continuous random variable is a type of variable that can take infinite values within a given range. In the context of the exercise, the variable \(x\) is continuous, meaning it can assume any value between a lower and upper limit, unlike a discrete random variable that has countable outcomes.

This distinction is important because it determines how we calculate probabilities. For continuous random variables, probabilities are found for ranges of values rather than for specific, exact numbers. This probability is represented as the "area under the curve" of the distribution graph, also known as the normal distribution curve.

When dealing with continuous random variables, it is essential to use probability density functions. These functions help you find the likelihood of a variable falling within a particular range.

The Z-Score is a statistic that measures how many standard deviations an element is from the mean of the distribution. For example, in our exercise problem, Z-scores help determine how far the values 29, 36, 22, and 35 are from the mean of 25 in standard deviation units.

The Z-Score is calculated using the formula:

• \( Z = \frac{X-\mu}{\sigma} \)

Where \(X\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

This measurement brings all data to the same scale, allowing comparison across different sets. In this exercise, calculating the Z-scores for given values transforms them into the standard normal distribution, a pivotal step for finding probabilities.

Standard deviation is a statistical measure that shows the amount of variance or dispersion in a set of values. A smaller standard deviation indicates that the data points tend to be closer to the mean, while a larger standard deviation signifies that they spread out over a broader range.

In our exercise, the standard deviation is given as 6, which means that the values of \(x\) typically deviate, or stray, 6 units from the average which is 25. This helps in understanding the spread of the distribution and in calculating the Z-Score more accurately.

Understanding standard deviation is crucial because it plays a fundamental role in the equation to transform values into Z-scores. It gives a standardized way of looking at how spread out or clustered together data is relative to the mean.

Probability in the context of a normal distribution and continuous random variables involves finding the area under the curve for a given range. In normal distribution, this curve is bell-shaped, symmetric about its mean.

To find the probability of \(x\) being between two points, you need the Z-scores for those points. These Z-scores are then looked up in a standard normal probability table, which provides the area under the curve to the corresponding Z-score.

For example, if you want to find the probability that \(x\) is between 29 and 36, and the Z-scores are 0.67 and 1.83, respectively, you consult the probability table to find the areas under the curve and subtract them. This difference further reveals the probability of the distributed values lying between those two bounds.

Probabilities from such calculations are fundamental in statistics as they provide insights into the likelihood of events occurring within specified ranges.