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When dealing with normal distributions, you'll often need to convert your raw data points into z-scores. A z-score represents how many standard deviations an element is from the mean. The formula for finding a z-score is given by \(z = \frac{x - \mu}{\sigma}\), where \(x\) is the data point, \(\mu\) is the mean of the dataset, and \(\sigma\) is the standard deviation.

This transformation helps us understand where our particular data point fits in a normal distribution. Once transformed into z-scores, you can use these values to look up probabilities in a standard normal distribution table. This table shows the area under the curve to the left of a particular z-score, giving the likelihood that the value of a random variable is less than your specified value. For instance, in solving the given exercise, converting our limits of 40 and 47 into z-scores helps us identify segments of the data's distribution.

In practice, these z-score calculations allow researchers, scientists, and statisticians to initially assess where data points stand in relation to the rest of a dataset or in comparison to a theoretical population. It offers a universal way to compare and interpret data.

Probability allows us to quantify the likelihood of an event happening. In the context of a normal distribution, when you're asked for a probability that a certain value falls between two points, you're essentially looking for the area under the distribution curve that lies between those values.

In the given problem, we calculated the probability that the variable \(x\) falls between 40 and 47 for a normally distributed data set with a mean of 50 and a standard deviation of 15. We achieved this by converting these values into their corresponding z-scores and using the standard normal distribution table to find the area under the curve.

The beauty of using probability with normal distributions is its standardization. By converting any normal distribution into a standard form using z-scores, you only need a single standard normal distribution table to find probabilities. This simplifies the process significantly, making it a critical tool for statistical analysis.

Standard deviation is a measure of how spread out the numbers in a data set are around the mean. When we say that a dataset has a larger standard deviation, it means the data is more spread out, and when it's smaller, the data points tend to be closer to the mean.

In our exercise, the standard deviation was \(\sigma = 15\). This tells us about the variability of our normal distribution. When calculating probabilities or converting raw scores to standardized scores, this value helps adjust for the spread of data points.

Understanding standard deviation is fundamental because it provides insight into the reliability and precision of the mean as the central value. A small standard deviation indicates that observations are closely clustered around the mean, suggesting consistent data. Conversely, a larger standard deviation indicates observations are more spread out, which can suggest more variability in the data.