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The Z-score is a valuable statistical tool used to evaluate where a specific data point lies in relation to the mean of a data set. It helps in determining how many standard deviations a particular point is from the mean. In the context of a normal distribution, the Z-score is calculated using the formula:\[Z = \frac{X - \mu}{\sigma}\]Where:

• \(X\) is the value for which we are calculating the Z-score.
• \(\mu\) is the mean of the data set.
• \(\sigma\) is the standard deviation.

By computing the Z-scores for the lower and upper height limits in our height distribution problem, \(Z_{150} = -2.22\) for \(150\, \text{cm}\) and \(Z_{195} = 2.78\) for \(195\, \text{cm}\), you're identifying how extreme these values are and using them to find the associated probabilities from a standard normal distribution table.

Standard deviation offers insight into the spread or dispersion of a data set. This statistical measure reveals how much members of a group deviate from the average value. Lower standard deviation means that the data points tend to be close to the mean, while a higher standard deviation indicates more variation among data points.For instance, with our example data where the mean height is \(170\,\text{cm}\) and the standard deviation is \(9\,\text{cm}\), the standard deviation indicates that the heights are generally \(9\,\text{cm}\) away from the mean. Standard deviation is crucial as it effects the Z-score calculation. It shows how typical or atypical a height is relative to 500 given heights in the example.To visualize it, consider a bell curve (the shape of a normal distribution), where nearly 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.

Cumulative probability is fundamental in the concept of probability distributions. It calculates the likelihood that a random variable will assume a value less than or equal to a specific value. By using the Z-scores and finding cumulative probabilities from a Z-table (or standard normal distribution table), one can determine the percentage of data lying within a certain range.In our problem, the cumulative probability for \(Z = -2.22\) is about \(0.0132\). This means approximately 1.32% of people are expected to have heights less than \(150\,\text{cm}\). Similarly, for \(Z = 2.78\), the cumulative probability is about \(0.9973\), suggesting that roughly 99.73% of people have heights less than \(195\,\text{cm}\).To find the probability of having a height between \(150\,\text{cm}\) and \(195\,\text{cm}\), the cumulative probability for \(Z = -2.22\) is subtracted from that for \(Z = 2.78\). Hence, \(0.9973 - 0.0132 = 0.9841\), suggesting that 98.41% of people will likely fall within this height range, translating to approximately 492 individuals among 500.