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Understanding the z-score is crucial in statistics, as it provides a way to measure the distance between a particular value and the population mean in units of the standard deviation. In other words, it tells us how many standard deviations an element is from the mean.

Let's break down the steps of calculating a z-score using the example from the exercise: we have college women's heights, with a mean of 65 inches and a standard deviation of 2.5 inches. To find the z-score for a woman who is 67 inches tall, we subtract the mean from the observed value, 67 inches, and divide by the standard deviation. The formula for this calculation looks like this:
\[ z = \frac{(X - \mu)}{\sigma} \]
By plugging in the values, we get:
\[ z = \frac{(67 - 65)}{2.5} = 0.8 \]
This result tells us the woman's height is 0.8 standard deviations above the mean. This step is a fundamental part of statistical data analysis because it standardizes different data points, allowing for comparison across different distributions or datasets.

Once we have the z-score, the next step is to interpret this score. This is where the standard normal distribution table, often referred to as the z-table, comes into play. This table provides the probability or area under the curve to the left of a specified z-score in a standard normal distribution. The standard normal distribution is a special case of the normal distribution that has a mean of 0 and a standard deviation of 1.

To consult the table, one would find the row for the first two digits of the z-score and the column corresponding to the second decimal place, if available. In the case of a z-score of 0.8, the table gives us a value of 0.7881. This number represents the probability that a randomly selected woman's height is less than 67 inches, telling us that roughly 78.81% of college women fall below that height.

Indeed, the z-table is an invaluable tool for statisticians and analysts alike, letting them ascertain probabilities and percentiles without complex calculations. If improvement is needed, practice using the z-table with varying scores to build familiarity and fluency.

The final stage in our process is the application of our findings from the z-score and z-table to perform the data analysis. The goal is to make a real-world interpretation of the statistical results. In our exercise, we needed to find out what percentage of college women are taller than 67 inches.

We already discovered that about 78.81% are shorter, thanks to the z-table. To determine the remaining percentage, we simply subtract from 100%, as the total probability of any event (such as being taller or shorter than a given height) must equal 100%. This yields:
\[ 100\% - 78.81\% = 21.19\% \]
Thus, 21.19% of college women are taller than 67 inches. Through statistical data analysis, we transform abstract numbers into meaningful information, such as understanding height distributions, performance metrics, or financial risk assessments. Mastery of this practice is critical for students and professionals as it supports data-driven decision-making processes across various fields.