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When dealing with normal distributions, one of the first and crucial steps is calculating the Z-score. It serves as a way to transform your data into a standardized form, which aids in comparing different points in a distribution.
To compute a Z-score, we employ the formula: \[ z = \frac{x - \mu}{\sigma} \]where:

• \(x\) - the value in the distribution you're examining
• \(\mu\) - the mean of the distribution
• \(\sigma\) - the standard deviation

By converting raw data points into Z-scores, you center your data around a mean of zero with a standard deviation of one. This helps to simplify probability calculations, as we'll see in the following sections. Given the problem where \(\mu = 4\) and \(\sigma = 2\), calculating the Z-scores for bounds at \(x = 3\) and \(x = 6\) gives us \(z = -0.5\) and \(z = 1.0\), respectively. These Z-scores are used to find probabilities within the standard normal distribution.

The standard normal distribution, often referred to as the Z-distribution, plays a key role in statistics. It is a normal distribution with a mean \(\mu\) of 0 and a standard deviation \(\sigma\) of 1.
This is a special form of normal distribution that allows us to map any normal distribution to it using Z-scores. Essentially, when we convert our data using Z-scores, we're mapping it onto this template standard distribution. This transformation normalizes any dataset, facilitating easier probability analysis.
Most statistical tables, such as the Z-table, are designed based on this standard normal distribution. By converting different datasets into Z-scores, these tables can be universally applied to find probabilities. In practical terms, once the Z-scores of \(-0.5\) and \(1.0\) are determined, they can be used to look up precise probabilities directly from the Z-table.

Calculating the probability for a specific range in a normal distribution involves several steps, starting from transforming your values to a standard form and concluding with range probability analysis.
Once Z-scores are determined, these serve as inputs to ascertain cumulative probabilities from a Z-table. For example:

• The Z-score of \(-0.5\) corresponds to a cumulative probability roughly equal to \(0.3085\)
• A Z-score of \(1.0\) holds a cumulative probability close to \(0.8413\)

To find the probability that \(x\) falls between 3 and 6, these individual cumulative probabilities are leveraged. Specifically, on a range, this is done by subtracting the lower cumulative probability from the higher one. Hence, the calculation is expresssed as: \[P(3 \leq x \leq 6) = P(z \leq 1.0) - P(z \leq -0.5)\]This results in \(0.8413 - 0.3085 = 0.5328\). Thus, the probability that \(x\) will fall between the values of 3 and 6 in this distribution is approximately \(53.28\)\%. This makes it clear and straightforward to understand how probabilities manifest across ranges within normal distributions.