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The Z-score is a numerical measurement that describes a value's relation to the mean of a group of values. To calculate 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.

A Z-score tells you how many standard deviations an element is from the mean. A Z-score of 0 indicates the value is exactly at the mean. Positive Z-scores represent values above the mean, while negative Z-scores represent values below the mean.
In the context of a normal distribution, Z-scores help us find probabilities and identify percentile ranks. This is useful for various tasks, such as determining what proportion of data falls below a certain point in a normal distribution. Understanding how to convert areas into Z-scores and then into actual values of \( x \) is crucial when working with normal distributions.

A probability density function (PDF) is a function that describes the likelihood of a random variable to take on a given value. For a continuous random variable, the PDF assigns a probability to all outcomes in the space of the random variable, helping us understand the distribution.
The normal distribution, often known as the Gaussian distribution, is one of the most frequently used probability distributions in statistics due to its excellent naturally occurring properties. Its PDF is described mathematically by the equation: \[ f(x | \mu, \sigma) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(x - \mu)^2}{2 \sigma^2}} \]

• \( \mu \) is the mean.
• \( \sigma \) is the standard deviation.

This bell-shaped curve shows that data near the mean are more frequent in occurrence than data far from the mean.
The PDF not only helps in understanding the data but is instrumental for other statistical calculations and analysis, especially when working with the normal distribution and the normal curve behaviors.

The area under the curve of a probability density function, in the context of the normal distribution, represents the probability of a random variable falling within a particular range. The total area under the standard normal distribution curve is always equal to 1, which represents the total probability of all possible outcomes.
When dealing with normal distributions, finding the area to the left or right of a certain point can give us the cumulative probability up to that point. This is why it is so useful to convert percentages or probabilities into Z-scores.
To find specific values, like in our exercise with questions a to f, you would:

• Use a Z-score table or calculator to find the cumulative probability.
• Apply the Z-score formula to determine the value of \( x \).

These tools are crucial to solve practical problems involving normal distributions, such as determining threshold values in quality control or marking grades on a curve. Grasping the concept of area under the curve not only aids in understanding distributions but also in making informed predictions grounded in statistical analysis.