91Ó°ÊÓ

In statistics, a normal distribution is a fundamental concept. It is often called a 'bell curve' because of its bell-shaped appearance. Many naturally occurring data sets, like heights, test scores, and lengths, follow this distribution pattern. A normal distribution is symmetrical around its mean, which means most data points cluster around the middle, and fewer lie at the extremes.
In the context of the exercise, a male’s back-to-knee length is normally distributed with a mean of 23.5 inches and a standard deviation of 1.1 inches. This symmetry allows us to calculate probabilities for different ranges using the properties of the normal curve.

A Z-score is a statistical measurement that describes a value's relation to the mean of a group of values. It is measured in terms of standard deviations from the mean. In other words, it tells you how many standard deviations away a value is from the average. The formula for calculating the Z-score is:
\[ Z = \frac{X - \text{mean}}{\text{standard deviation}} \]
For example, in the exercise, we converted the back-to-knee lengths of 22.0 inches and 24.0 inches to their respective Z-scores:
For 22.0 inches:
\[ Z_{\text{lower}} = \frac{22.0 - 23.5}{1.1} = -1.36 \]
For 24.0 inches:
\[ Z_{\text{upper}} = \frac{24.0 - 23.5}{1.1} = 0.45 \]
Interpreting these Z-scores, -1.36 means 22.0 inches is 1.36 standard deviations below the mean, and 0.45 means 24.0 inches is 0.45 standard deviations above the mean.

The Standard Normal Distribution Table, or Z-table, is a mathematical table used to find the probability that a statistic is less than or equal to a given Z-score. This table is essential for finding probabilities when dealing with normal distributions.
Once you have calculated the Z-scores, you can refer to the Z-table to find the corresponding probabilities. For instance, in the exercise:

• The Z-score for the lower bound (22.0 inches) is -1.36, which corresponds to a probability of approximately 0.0869.
• The Z-score for the upper bound (24.0 inches) is 0.45, which corresponds to a probability of approximately 0.6736.

By using these values, you can determine the probability that a value lies between two points on a normal distribution.

Probability calculation in the context of a normal distribution often involves converting the range of interest into Z-scores and then finding the relevant probabilities using the Z-table.
In the provided exercise, we are asked to find the probability that a male has a back-to-knee length between 22.0 inches and 24.0 inches. We do this in steps:

• First, convert the given values (22.0 inches and 24.0 inches) into Z-scores.
• Use the Z-table to find the probabilities that correspond to these Z-scores. For this exercise, the probabilities are 0.0869 and 0.6736 for Z-scores of -1.36 and 0.45, respectively.

Finally, subtract the smaller probability from the larger one to find the probability of the value lying in the specified range:
\[ P(22.0 < X < 24.0) = 0.6736 - 0.0869 = 0.5867 \]
This tells us that there is approximately a 58.67% chance that a male's back-to-knee length falls between 22.0 inches and 24.0 inches.