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When we talk about women's height in North America, we're dealing with a set of data that often forms a normal distribution. This is a common way of representing data in statistics because it showcases how most women will be around an average height, with fewer women being significantly taller or shorter.
The average height, or mean, in this case, is 65 inches, and the standard deviation is 3.5 inches.
Understanding these two measures is essential because they help us comprehend how much variation there is from the average height.

• The average, \( \mu \), is the typical height that you would expect.
• The standard deviation, \( \sigma \), provides insights into the spread of heights.
  A small \( \sigma \) means most heights are clustered around the average, while a larger \( \sigma \) means more variation.

This normal distribution becomes a crucial factor when determining how many women meet specific criteria, like a height requirement for certain jobs.

To find out how many women fall below the minimum height requirement, we use something known as a Z-score. A Z-score tells us how many standard deviations away a particular value is from the mean. This is helpful for assessing where a certain measurement falls in a distribution.
To calculate the Z-score, we use the formula: \[ Z = \frac{X - \mu}{\sigma} \], where:

• \( X = 62 \) inches (minimum height requirement).
• \( \mu = 65 \) inches (average height).
• \( \sigma = 3.5 \) inches (standard deviation).

Plugging the numbers into the formula gives us a Z-score of approximately \( -0.857 \). This implies that 62 inches is \( 0.857 \) standard deviations below the mean, indicating how this particular height compares to the average.

The concept of a standard normal distribution is fundamental to statistics and useful in many real-life applications. In this context, the standard normal distribution is a special normal distribution with a mean of 0 and a standard deviation of 1. Instead of measuring absolute values like height, it measures Z-scores, which reflect how data compares to the average. This allows us to compare diverse datasets on a standard scale.By standardizing the data to a Z-score, we can refer to a Z-table to find probabilities and proportions related to any Z-score value.

• The Z-table helps translate the Z-score back to an understandable proportion of data.
• For instance, in our context, the Z-score of \( -0.857 \) is used to determine the proportion of women who are shorter than the minimum height requirement.

Utilizing the standard normal distribution simplifies the process of understanding where a particular measurement stands within a population.

Cumulative probability is a crucial concept when dealing with normal distributions and Z-scores. It refers to the probability that a random variable is less than or equal to a certain value. This concept is directly applicable when figuring out how many women do not meet a specific height threshold.
When you calculate a Z-score, as we did earlier, you can then reference a Z-table to find the cumulative probability. The Z-table shows the probability of a given Z-score or less. In our scenario, a Z-score of \(-0.857\) corresponds to a cumulative probability of approximately 0.1957.

• This means about 19.57% of women are below the height requirement.
• A cumulative probability near this number suggests that only a small portion of females fall short of the height needed to be a flight attendant.
• It also indicates how commonly such values occur, helping organizations set fair thresholds.

Understanding cumulative probability aids significantly in making informed decisions and predictions based on statistical data.