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Bone density tests measure the density of minerals in bones, indicating bone strength and health. These scores follow a normal distribution with a mean of 0 and a standard deviation of 1.
The normal distribution concept is like a bell curve, where most test scores are around the mean, and fewer scores are far from it. By knowing the mean and standard deviation, we can predict how likely a specific score is.
In our example, we're interested in scores between 1.50 and 2.50.

Z-scores tell us how many standard deviations a data point is from the mean. When the data is normally distributed with a mean of 0 and a standard deviation of 1, the Z-score for any value is simply that value itself.
For instance, a Z-score of 1.50 means the score is 1.50 standard deviations above the mean. Similarly, a Z-score of 2.50 is 2.50 standard deviations above the mean.
Z-scores help us to compare data points from different distributions and find probabilities related to those points using standard normal distribution tables or technology.

Cumulative probability is the probability that a random variable falls within a specified range. For normal distributions, we use the cumulative distribution function (CDF) to find this.
The CDF value at a specific Z-score tells us the total probability for scores up to that Z-score.
For example, using a Z-table or statistical software, the CDF for Z = 1.50 is 0.9332, and for Z = 2.50, it is 0.9938.
To find the probability between two Z-scores, subtract the lower cumulative probability from the higher one.
In this case, the probability of bone density scores being between 1.50 and 2.50 is: \(0.9938 - 0.9332 = 0.0606\). This means there's a 6.06% chance a score will fall in that range.