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The standard normal distribution is a cornerstone of statistics. It's essentially a normal distribution that has been standardized to have a mean (bc) of 0 and a standard deviation (c3) of 1. By converting any normal distribution to a standard one, we can easily compare different distributions and calculate probabilities.
This transformation frees us from the need to handle complex individual distributions in their original forms. Instead, we have a universal model to reference.

• Standard normal distributions are key to understanding probability because they allow use of the standard normal table.
• It provides a way to determine how usual or unusual a particular data point is from the mean.

This simplification is achieved through the concept of the Z-score, which we will examine next.

The Z-score is a numerical measurement used to express how far a particular data point is from the mean, measured in units of standard deviation.
To calculate the Z-score, you use the formula:\[z = \frac{x - \mu}{\sigma}\]where:

• \(x\) is the value of the data point,
• \(\mu\) is the mean of the data set,
• \(\sigma\) is the standard deviation of the data set.

In our exercise, we calculated the Z-score for \(x = 30\), with \(\mu = 20\) and \(\sigma = 3.4\), and found it to be 2.94.
This means that a value of 30 is 2.94 standard deviations above the mean.

• The Z-score helps us assess whether a data point lies within a common or uncommon range when compared to our mean.
• It simplifies the process of probability calculation, enabling the quick lookup of ranges in the standard normal distribution table.

Calculating probability in the context of a normal distribution involves determining the likelihood of a random variable falling within a given range.
We use the Z-score to achieve this by transforming the original distribution problem into a problem of the standard normal distribution.
Once we have our Z-score, we can use the standard normal distribution table to find probabilities.

Looking Up Table Values

Most standard normal distribution tables give the cumulative probability from the mean up to a Z-score.
This means that when we find \(P(z \leq 2.94)\), it represents all the area to the left under the curve up to 2.94 standard deviations.

• In our case, \(P(z \leq 2.94)\) is 0.9984, representing about 99.84% of the data.
• To find \(P(z \geq 2.94)\), which corresponds to \(x \geq 30\), we simply calculate the complement: \(1 - 0.9984 = 0.0016\).

This indicates there's a very small probability (0.16%) that a randomly chosen value is greater than or equal to 30.
The ability to convert probabilities through the use of Z-scores and tables makes probability calculations straightforward and consistent.