91Ó°ÊÓ

In statistics, the mean is the average of a set of numbers, in this case, it's the average height of men which is 69 inches. The mean serves as a central point around which the data is distributed. The standard deviation tells us how spread out the numbers are from this average. Here, the standard deviation is 3 inches, indicating that most men's heights will fall within 3 inches above or below 69 inches.

The mean and standard deviation are crucial for understanding the normal distribution, especially when applying it to real-world data such as heights. It's important to realize that in a normal distribution, the majority of data points will lie close to the mean. In this exercise, these values help us calculate how unusual or usual a particular height is by using another concept known as the Z-score.

A Z-score indicates how many standard deviations an element is from the mean. It is calculated using the formula: \[ Z = \frac{(X - \text{Mean})}{\text{Standard Deviation}} \] where \(X\) is the actual value in the data set.

In this problem, we're asked to find heights corresponding to specific Z-scores. For a Z-score of 2.00, we calculate the height as: 68 inches + 2 * 3 inches = 75 inches.

For a Z-score of -1.50, the calculation is: 69 inches - 1.5 * 3 inches = 64.5 inches.

This means a person 75 inches tall is 2 standard deviations above the average height, while one 64.5 inches tall is 1.5 standard deviations below. Knowing Z-scores helps us quickly see how far an individual's measurement is from the norm and in which direction.

A symmetric distribution means that the data is evenly distributed around the mean. This implies that the shape of the distribution is mirrored around the central point, which in this context is the mean height of 69 inches.

In the case of men's heights, this symmetry means there are just as many men above the average height as there are below. The normal distribution, often represented as a bell-shaped curve, is a common example of a symmetric distribution. It helps us understand that extreme values (like very tall or very short individuals) are less common than those around the average.

This symmetry, when paired with known values of mean and standard deviation, helps statisticians and researchers make predictions about data and understand variability within the population.