Question

In Europe, \(8 \%\) of men are colorblind. \(^{20}\) Consider taking repeated samples of 20 European men. (a) What is the mean number of colorblind men? (b) What is the standard deviation of the number of colorblind men?

Short answer

The mean number of colorblind men is 1.6, and the standard deviation is approximately 1.213.

Step-by-step solution

Identifying the Distribution

The problem can be treated as a binomial distribution problem because there are a fixed number of independent trials (20 men), each with two possible outcomes (colorblind or not colorblind), and a constant probability of success (being colorblind) for each trial.

Calculating the Mean

For a binomial distribution, the mean, or expected value, is calculated by multiplying the number of trials by the probability of success on a single trial: \( \mu = n \times p \), where \( n \) is the number of trials, and \( p \) is the probability of success. Here, the number of trials \( n = 20 \), and the probability of success \( p = 0.08 \). Thus, the mean is \( \mu = 20 \times 0.08 = 1.6 \).

Calculating the Standard Deviation

The standard deviation of a binomial distribution is given by the formula \( \sigma = \sqrt{n \times p \times (1 - p)} \), where \( n \) is the number of trials, \( p \) is the probability of success, and \( 1 - p \) is the probability of failure. Using the values given, the standard deviation is \( \sigma = \sqrt{20 \times 0.08 \times (1 - 0.08)} = \sqrt{20 \times 0.08 \times 0.92} = \sqrt{1.472} \approx 1.213 \).

Key concepts

Mean of a Binomial Distribution

Understanding the mean of a binomial distribution is central to grasping the expected outcome of a set of identical and independent trials, where each trial can only result in success or failure. In the context of our exercise, where we're examining colorblindness among European men, the mean tells us the average number of colorblind men we can expect in samples of 20.

To calculate the mean \( \mu \), we multiply the total number of trials \( n \) by the probability of securing a 'success' in a single trial \( p \). Here, success is defined as encountering a man who is colorblind. With \( n = 20 \) samples and the probability \( p = 0.08 \) (or 8%), the mean number of colorblind men per sample of 20 is \( \mu = n \times p = 20 \times 0.08 = 1.6 \).

This value signifies that in any random group of 20 European men, on average, approximately 1.6 of them will be colorblind.

Standard Deviation of a Binomial Distribution

The standard deviation of a binomial distribution, denoted as \( \sigma \), provides a measure of variation or dispersion from the mean. A lower standard deviation indicates that the data points tend to be closer to the mean, while a higher standard deviation indicates that the data are more spread out. For the scenario with the colorblind European men, this statistic helps us understand the extent of fluctuation we might expect around our previously calculated mean value of 1.6 colorblind individuals.

To compute the standard deviation, you apply the formula \( \sigma = \sqrt{n \times p \times (1 - p)} \). From our prior example, we have \( n = 20 \) trials and a probability of success \( p = 0.08 \) for each trial. Plugging these into our formula, we get \( \sigma = \sqrt{20 \times 0.08 \times (1 - 0.08)} = \sqrt{1.472} \approx 1.213 \).

An approximate standard deviation of 1.213 suggests that the number of colorblind men in different sample groups of 20 will typically deviate about 1.213 from the average of 1.6, some samples might have none, and others could have two or three colorblind individuals.

Probability of Success in Binomial Distribution

In binomial distribution problems, the term 'probability of success' refers to the likelihood of a particular outcome we're looking to track. It's crucial to note that 'success' doesn't imply something positive; it's simply the outcome of interest. In our case, the probability of success \( p \) is the chance that a randomly selected European man is colorblind, which has been provided as 8%, or 0.08.

The probability of success remains constant across all trials, meaning each man has the same 8% chance of being colorblind, irrespective of previous or future trials, due to the assumption of independence in binomial trials. This constant probability is what allows us to use the binomial distribution model effectively, as it creates a predictable pattern that can be analyzed statistically. Understanding this probability is the foundation of determining the mean, standard deviation, and indeed all other statistical measures that emerge from the binomial distribution framework.