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Calculating a Z-score is essential when you want to see how a specific value compares to the rest of the data in a normal distribution. The formula to find the Z-score is:- \(Z = \frac{x - \mu}{\sigma}\) - \(x\) is the value you are examining. - \(\mu\) is the mean of the dataset. - \(\sigma\) is the standard deviation.This formula helps you identify where your specific value lies in relation to the mean. For example, in this exercise, to determine how a height of 67 inches (5 ft 7 in) fits in the distribution of female heights, you calculate the Z-score. By substituting the values into the equation \( (x = 67, \mu = 66, \sigma = 2) \), you get a Z-score of 0.5. This means the height is half a standard deviation above the mean.Understanding the Z-score helps you analyze whether a claim about a population value, like the minimum height required, is valid based on its statistical position.

The normal distribution, often called the bell curve, is a fundamental concept in statistics. It's symmetric, showing that most outcomes are near the mean, reflecting a balanced spread of data.Key properties include:- **Symmetric Shape**: The left and right sides of the graph are mirrored.- **Mean, Median, Mode Alignment**: The mean, median, and mode are identical and located at the center of the distribution.- **Standard Deviation Impact**: Determines the spread of the data; 68% of data lie within one standard deviation (±1\(\sigma\)), and about 95% within two standard deviations (±2\(\sigma\)).In this problem, knowing these properties helps in understanding how the heights of people spread around the average height of 66 inches. This tells us most women's heights are close to this mean, and using the standard deviation, we can gauge how common or rare certain heights are, such as 67 inches.

The proportion of a normal distribution associated with a Z-score represents the percentage of data below or above a particular point. This is also known as the area under the curve. Here's how it works: - **Finding Probability**: By referring to a Z-table or using statistical software, we can find the probability related to a specific Z-score. This reflects the proportion of the dataset expected to fall below a given point. In our exercise, we wanted to determine what proportion of women are shorter than 67 inches and thus not excluded by the height restriction. Given the calculated Z-score of 0.5, looking up this score shows about 69.15% fall below this height. Thus, only this proportion of women meet the condition, contradicting the 94% claim made in the original problem statement. Understanding this principle helps evaluate claims about cut-off points in datasets and how such restrictions might unfairly target a segment of the population.