Problem 108 - An Introduction to Mathematical Statistics and Its Applications

Question

Ten equally qualified applicants, six men and four women, apply for three lab technician positions. Unable to justify choosing any of the applicants over all the others, the personnel director decides to select the three at random. Let \(X\) denote the number of men hired. Compute the standard deviation of \(X\).

Step-by-Step Solution

Answer

The standard deviation of X is \( \sqrt{0.72} \).

Step 1: Identify the probability distribution

Given that there are equally qualified numbers of men and women applying for the positions and the hires are made randomly. This is a binomial distribution problem because there are only two possible outcomes - a hired technician is either a man or not a man.

Step 2: Calculate the mean for the binomial distribution

The mean for a binomial distribution is given by \( \mu = np \), where \( n \) is the number of trials and \( p \) is the probability of success. Here, \( n = 3 \) (the number of available positions) and \( p = 0.6 \) (probability of a man being hired, as there are 6 men among the 10 applicants). Hence, \( \mu = 3 * 0.6 = 1.8 \).

Step 3: Calculate the variance for the binomial distribution

The variance for a binomial distribution is given by \( \sigma^2 = np(1-p) \). Using the same \( n \) and \( p \) as the previous step, \( \sigma^2 = 3 * 0.6 * 0.4 = 0.72 \).

Step 4: Calculate the standard deviation

The standard deviation is the square root of the variance. So, \( \sigma = \sqrt{\sigma^2} = \sqrt{0.72} \).

Key Concepts

Understanding Standard Deviation

Standard deviation is a key concept in statistics, offering insight into the variability or spread of a set of data. In simpler terms, it helps us understand how much the data points differ from the mean.

If the data points are close to the mean, the standard deviation is small; if they are spread out widely, it is large. Mathematically, standard deviation is represented as \( \sigma \) and calculated as the square root of the variance.

• To find the standard deviation in a binomial distribution, compute the variance using \( np(1-p) \), where \( n \) is the number of trials and \( p \) is the probability of success.
• In our exercise, variance \( \sigma^2 \) equals \( 0.72 \).
• The standard deviation is then \( \sigma = \sqrt{0.72} \).

This gives us a precise measure of how much the outcomes deviate from the expected number of men hired.

Exploring Probability Distribution

The probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In a binomial distribution, there's a fixed number of trials, two possible outcomes, and a constant probability of success.

In the original problem, the distribution is binomial because it involves hiring men (success) or not (failure):

• Trials: The number of hires, \( n = 3 \).
• Probability of success: Probability of hiring a man, \( p = 0.6 \).

This distribution allows us to calculate probabilities for different numbers of men being hired out of the three positions, making it a useful tool for predicting outcomes in events with similar characteristics.

Grasping Variance

Variance helps us understand the spread or dispersion of a set of probability outcomes around the mean. It is denoted by \( \sigma^2 \) and provides a square measure of how far each data point is from the mean.

For a binomial distribution, the variance is calculated as \( np(1-p) \):

• \( n = 3 \) trials or hires.
• \( p = 0.6 \), the probability of hiring a man.
• Variance \( \sigma^2 = 3 \times 0.6 \times 0.4 = 0.72 \).

By knowing the variance, we can better grasp how much the number of men hired is likely to differ from the expected mean, providing a more comprehensive view of risk and variability.