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A Z-score helps us understand how far away a certain value is from the mean in a data set. In simpler terms, it tells us how many "steps" or standard deviations a data point is from the average. To calculate the Z-score, we use the formula:

• Subtract the mean (\(\mu\)) from the value in question (\(X\)).
• Then, divide by the standard deviation (\(\sigma\)).

For example, if we are looking for the Z-score of a woman's height of 67 inches, given that the average height is 65 inches with a standard deviation of 2.5 inches, we proceed as follows:

• First, find the difference: \(67 - 65 = 2\).
• Next, divide by the standard deviation: \(2/2.5 = 0.8\).

Thus, the Z-score is \(0.8\), indicating this height is \(0.8\) standard deviations above the mean. This Z-score becomes a powerful tool in determining how unusual or typical a certain observation is within a normal distribution.

A probability distribution describes how the values of a random variable are spread or distributed. For a normally distributed variable, like heights in this scenario, the probability distribution is symmetrically centered around the mean. The shape of this distribution is often referred to as a "bell curve." Think about it as a way to visualize where most observations fall.
With a normal distribution, you can find probabilities related to a specific range of outcomes. In our example, once the Z-score for a height of 67 inches is calculated, the next step is to understand what this score means in terms of probability:

• The Z-score of \(0.8\) indicates that the height is above average.
• Using a Z-table, or statistical software, you can find the cumulative probability up to that Z-score, which is \(0.7881\) in this case.
• This means there's a \(78.81\%\) chance of encountering a height less than 67 inches.

To find out the probability of heights greater than 67 inches, simply subtract from 1 (since total probability sums to 1) giving: \(1 - 0.7881 = 0.2119\) or \(21.2\%\). This shows that approximately \(21.2\%\) of college women are taller than 67 inches based on this probability distribution.

StatCrunch is a statistical software tool that helps you perform data analysis and view probability distributions quickly. It's like a calculator but with more capabilities, tailored to handling statistical tasks.
When dealing with normal distributions like this one, StatCrunch can be used to determine the probability or percentile related to a specific Z-score. Here’s why StatCrunch is useful:

• It eliminates the need to manually search through Z-tables.
• The software quickly computes cumulative probabilities, helping find the percentage of data that falls under or above a certain point.
• StatCrunch visualizes results with graphs that help in understanding the distribution.

Using StatCrunch for calculating the height distribution in this exercise:

• The Z-score of \(0.8\) was input into StatCrunch.
• The software returned a probability of \(0.7881\), representing those shorter than 67 inches.
• By subtracting from 1, the tool efficiently gave the percentage taller than 67 inches as \(21.2\%\).

This makes StatCrunch particularly handy in academic tasks, providing accuracy effortlessly, and allowing you to focus on understanding data rather than getting tangled in calculations.