Problem 5 - An Introduction to Psychological Statistics

Question

The heights of men in the United States are normally distributed with a mean of 69.1 inches and a standard deviation of 2.9 inches. What proportion of men are taller than 6 feet (72 inches)?

Step-by-Step Solution

Answer

Approximately 15.87% of men are taller than 72 inches.

Step 1: Convert Height to Z-Score

To find the proportion of men taller than 72 inches, we first convert the height into a z-score using the formula: \[ z = \frac{X - \mu}{\sigma} \] where \(X = 72\) inches (height we are considering), \(\mu = 69.1\) inches (mean), and \(\sigma = 2.9\) inches (standard deviation). Substituting the values we have: \[ z = \frac{72 - 69.1}{2.9} \approx 1.00 \].

Step 2: Find the Proportion Using the Z-Table

Use the z-score to find the corresponding probability in a standard normal distribution table (z-table). A z-score of 1.00 corresponds to a cumulative probability of 0.8413. This represents the proportion of men shorter than 72 inches.

Step 3: Calculate Proportion Taller

To find the proportion of men taller than 72 inches, subtract the cumulative probability from 1: \[ 1 - 0.8413 = 0.1587 \]. This means approximately 15.87% of men are taller than 72 inches.

Key Concepts

Z-Score and Its Importance

The z-score is a crucial concept when working with normal distributions. It is a measure that describes a data point's relation to the mean of the data set. The z-score indicates how many standard deviations a particular value (or observation) is from the mean.
When calculating the z-score, you use the formula:

• \( z = \frac{X - \mu}{\sigma} \) where:
• \( X \) represents the value under consideration,
• \( \mu \) is the mean of the data set,
• \( \sigma \) is the standard deviation.

If the z-score is positive, the data point is above the mean; if it is negative, the data point is below the mean. In the context of our exercise, we are looking at how much taller than the average 72 inches is, compared to the average height of 69.1 inches. After calculating, the z-score comes out to be approximately 1.00.

Understanding Standard Deviation

Standard deviation is a key statistical measure that reflects the amount of variation or dispersion in a set of data points. It tells us how much the data points typically stray from the mean. This understanding is fundamental when dealing with normal distributions, as it provides insights into the spread of the data.
Standard deviation is calculated using:

• Find the mean (average) of the data set.
• Subtract the mean and square the result for each data point.
• Calculate the average of these squared differences.
• Take the square root of that average, which is the standard deviation.

In the exercise, a standard deviation of 2.9 inches tells us that the majority of men's heights are within 2.9 inches of the mean, 69.1 inches, which helps us understand the range within which most people fall.

Calculating the Mean

The mean, often referred to as the average, is the sum of a list of numbers divided by the number of items in the list. It provides a central value for the data set, representing the overall trend. In context, the mean height of men in the United States has been calculated to be 69.1 inches. This mean forms the baseline from which we measure deviations or variations in individual heights.
To calculate the mean yourself:

• Add up all the individual data points.
• Divide the total sum by the number of data points.

By calculating the mean, you're able to get an immediate sense of what typical values in a set of data look like, which in turn helps in comparing any particular value to determine how extreme or common it is. For our exercise, knowing that the mean is 69.1 inches allows us to determine that 72 inches, the point of interest, is bigger than average.