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In statistics, a normal distribution is a probability distribution that is symmetric around the mean. Most values cluster around a central region, with values tapering off as you move further away from the mean. This bell-curved shape is quite common in various natural phenomena.
For instance, if you were to measure the heights of a large group of adult women, quite often the data will form a bell-shaped curve when plotted. Most women will have a height close to the average, and fewer women will fall into the extremely short or tall categories.
When referring to the problem involving General Electric, the mean height of adult women is given as 66 inches, illustrating the central point of the distribution. The standard deviation, which is 2 inches, shows how spread out the heights are from the mean. Many phenomena, like this one, follow the normal distribution, making it a foundational concept in statistics.

A z-score tells us how many standard deviations a data point is from the mean. This is crucial when you want to find out how extreme or typical a particular value is within its distribution.
The formula to calculate a z-score is: \( z = \frac{(X - \mu)}{\sigma} \), where:

• \( X \) is the value in question (e.g., 67 inches in our exercise).
• \( \mu \) is the mean of the distribution (66 inches).
• \( \sigma \) is the standard deviation (2 inches).

In this exercise, the z-score of 67 inches is calculated to be 0.5, indicating that this height is half a standard deviation above the mean. This calculation is pivotal, as it provides a way to standardize different data points, allowing for comparison within the same dataset.

A probability distribution assigns probabilities to all possible outcomes of a random variable. For continuous data like height, this distribution is often modeled as a normal curve.
In our scenario, the height of women is approximately normally distributed. With the calculated z-score, we can use a standard normal distribution table, often referred to as a z-table, to determine probabilities associated with different heights.
The z-table helps us to find the probability that a random variable falls below a particular z-score. For instance, a z-score of 0.5 correlates to a probability of 0.6915, meaning there is a 69.15% chance that a randomly selected woman will have a height less than 67 inches. Understanding these probabilities is essential for making informed conclusions based on statistical data.

Hypothesis testing is a method used to make statistical inferences about a population parameter based on a sample. It involves an initial assumption, the null hypothesis, and looks to determine whether there is enough evidence to reject it.
For the case of the height requirement by General Electric, a hypothesis could be that the restriction truly excludes 94% of women. Using the calculations from the z-score and the z-table, we determine that only 69.15% of women are shorter than 67 inches, challenging the claim in the lawsuit.
Hypothesis testing involves comparing the observed probability against the assumed hypothesis to validate or dispute it. It's a systematic and quantifiable way to test claims and assumptions, providing clarity and objective conclusions based on data.