Question

Heights of Men in the US Heights of adult males in the US are approximately normally distributed with mean 70 inches \((5 \mathrm{ft} 10 \mathrm{in})\) and standard deviation 3 inches. (a) What proportion of US men are between \(5 \mathrm{ft}\) 8 in and \(6 \mathrm{ft}\) tall \((68\) and 72 inches, respectively)? (b) If a man is at the 10 th percentile in height, how tall is he?

Short answer

a) The proportion of US men between \(5 \mathrm{ft} 8 \mathrm{in}\) and \(6 \mathrm{ft}\) tall is roughly 0.47. b) A man in the 10th percentile height-wise is approximately 64.4 inches tall.

Step-by-step solution

Calculate the Z-scores for Part a

In order to find the proportion of men between \(5 \mathrm{ft}\) 8 in and \(6 \mathrm{ft}\) tall, Z-scores need to be calculated first. The Z-score is obtained by subtracting the mean from the value and dividing by the standard deviation, giving \(Z = \frac{x - \mu}{\sigma}\). Calculate the Z-scores for 68 inches and 72 inches.

Use Standard Normal Distribution Table for Part a

The next step is to find the proportion of men with heights in that range. For this, use the standard normal distribution table. This table shows the probability that a normally distributed random variable with mean 0 and standard deviation 1 is less than a given number. The Z-score found in Step 1 will be used to consult this table and find the proportions. Subtract the proportion corresponding to 68 inches from the proportion corresponding to 72 inches to find the proportion of men with heights in the desired range.

Use the Percentile and Z-Score Formula for Part b

For part b, use the percentile given and the Z-score formula to find the height of a man in the 10th percentile. The Z-score for the 10th percentile can be found from the standard normal distribution table. Set up the Z-score formula \(Z = \frac{x - \mu}{\sigma}\) and plug in the Z-score for the 10th percentile, the given mean and standard deviation to solve for x, which is the height in inches.

Key concepts

Z-scores

When exploring the world of statistics, particularly the normal distribution, Z-scores are incredibly helpful. A Z-score transforms a data point into a numerical measurement that tells us how many standard deviations that data point is from the mean. Think of it as a 'distance' measure in the realm of statistics.

For example, if an adult male in the US is 72 inches tall, the Z-score would quantify how unusual his height is compared to the average US male height. Here's how you calculate it: \( Z = \frac{x - \mu}{\sigma} \), where \(x\) is the observed value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. In the context of our height example, 70 inches is the mean height (\(\mu\)) and 3 inches is the standard deviation (\(\sigma\)).

If we have an observed height of 68 inches, the Z-score calculation would show: \( Z = \frac{68 - 70}{3} \), leading to a Z-score which helps to identify how this height stands in comparison to the rest of the population.

A higher Z-score means the data point is further from the mean, and a lower score indicates it's closer. Moreover, a negative Z-score signifies that the data point is below the mean, while a positive score indicates it's above the mean. Understanding Z-scores is crucial for various statistical analyses including hypothesis testing and confidence intervals.

Standard Normal Distribution Table

The standard normal distribution table, also known as the Z-table, is a vital tool used in statistics to determine the probability that a standard normal variable falls within a particular range. This table is essentially a reference chart providing percentages (or probabilities) corresponding to Z-scores.

To bring this concept into focus, let's consider our previous discussion on Z-scores. Once we've calculated the Z-score for a particular value, like a US male's height, we use this score to look up how common or rare his height is. Usually, the table only lists positive Z-scores because the normal distribution is symmetrical, and negative values mirror the positive ones.

In our height example, if a man's height is 72 inches with a Z-score calculated, we'd reference this Z-score within the table to find the percentage of men who are shorter. By doing so for different heights, we start to understand the distribution of heights and how individuals relate to the overall population distribution. Using the table makes this process straightforward and quick, an essential part of applying normal distribution to real-world data.

Percentiles in Statistics

Percentiles are yet another fundamental component in the study of statistics. They split a data set into 100 equal parts, and each part represents one percent of the data. In essence, if someone's height is at the 50th percentile, this person is taller than half of the group of interest.

In reference to the provided exercise, they enable us to pinpoint exactly how tall a man is if he is at, for instance, the 10th percentile for height. It means he is taller than 10% of US men's heights in this example. The Z-score that corresponds to the 10th percentile is looked up in the Z-table. After finding that score, we can work backwards using the Z-score formula \( Z = \frac{x - \mu}{\sigma} \) to determine his actual height.

This reveals the inherent connection between percentiles and Z-scores: percentiles give us a relative position within a data set, and Z-scores provide the standardized value from which we can find that percentile and vice versa. This connection is extremely useful in educational testing, health-related fields, and various other disciplines where comparative data analysis is required.