Problem 41 - Basic Statistics for Business & Economics

Question

A federal study reported that \(7.5 \%\) of the U.S. workforce has a drug problem. A drug enforcement official for the state of Indiana wished to investigate this statement. In her sample of 20 employed workers: a. How many would you expect to have a drug problem? What is the standard deviation? b. What is the likelihood that none of the workers sampled has a drug problem? c. What is the likelihood at least one has a drug problem?

Step-by-Step Solution

Answer

a) Expected: 1.5, SD: 1.2. b) Likelihood of none: 0.2181. c) Likelihood of at least one: 0.7819.

Step 1: Define the Parameters

We know that 7.5% of the U.S. workforce has a drug problem. This is our probability of success (having a drug problem) for one individual: \( p = 0.075 \). The number of trials is the sample size, \( n = 20 \).

Step 2: Calculate the Expected Number (Part a)

To find the expected number of employed workers with a drug problem, use the formula for the expected value in a binomial distribution: \( E(X) = n \times p \). Substitute the given values: \( E(X) = 20 \times 0.075 = 1.5 \).

Step 3: Calculate the Standard Deviation (Part a)

The standard deviation in a binomial distribution can be calculated using the formula: \( \sigma = \sqrt{n \times p \times (1 - p)} \). Substitute the values: \( \sigma = \sqrt{20 \times 0.075 \times (1 - 0.075)} \approx 1.2 \).

Step 4: Calculate the Probability of None Having a Drug Problem (Part b)

This is a binomial probability problem where we need to find \( P(X = 0) \). Use the formula: \( P(X = 0) = \binom{n}{0} p^0 (1-p)^{n} \). So, \( P(X = 0) = (1-0.075)^{20} = 0.925^{20} \approx 0.2181 \).

Step 5: Calculate the Probability of At Least One Having a Drug Problem (Part c)

The probability of at least one having a drug problem is complementary to none having a problem: \( P(X \geq 1) = 1 - P(X = 0) \). From Step 4, \( P(X \geq 1) = 1 - 0.2181 \approx 0.7819 \).

Key Concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the likelihood of events occurring. It provides the framework to measure or quantify uncertainty. In the context of the given exercise, probability theory allows us to determine the likelihood of workers in Indiana having a drug problem based on a sample size. Simply put, probability is expressed as a number between 0 and 1, where 0 indicates an impossible event, and 1 represents certainty. Probabilities can also be expressed as percentages, making them easier to visualize in practical situations. In this exercise, the probability of an individual having a drug problem is 7.5%, or 0.075 as a decimal. The analysis involves using this likelihood to determine outcomes related to a group or sample of workers. Binomial distribution, a key concept here, is useful when we're interested in the number of times an event occurs in a fixed number of trials. It helps us assess situations where the outcomes are binary (like having or not having a drug problem).

Expected Value

The expected value is a key concept in probability and statistics, often called the mean or average. It represents the long-term average outcome of a random event over numerous trials. For a binomial distribution, the expected value is calculated using the formula: \[ E(X) = n \times p \]where \( n \) is the number of trials, and \( p \) is the probability of success on each trial. In our exercise, you calculate how many people in the sample are expected to have a drug problem. With \( n = 20 \) and \( p = 0.075 \), the expected number is \( E(X) = 20 \times 0.075 = 1.5 \). This result indicates that, on average, 1.5 workers out of a sample of 20 would have a drug problem if the entire population follows the same probability. This prediction doesn't mean you can have half a person, but it suggests that in repeated samples like this, the average number would converge near 1.5.

Standard Deviation

Standard deviation measures how spread out the numbers are in a data set. In the context of a binomial distribution, it gives us an idea of the variability or dispersion from the expected value. The formula for standard deviation in a binomial distribution is:\[ \sigma = \sqrt{n \times p \times (1 - p)} \]Using our exercise values \( n = 20 \) and \( p = 0.075 \):\[ \sigma = \sqrt{20 \times 0.075 \times (1 - 0.075)} \approx 1.2 \]This value tells us how much the number of workers with a drug problem might vary from the expected number of 1.5 in different samples of 20 workers. A smaller standard deviation would mean the outcomes are more consistent with the expected value, while a larger one indicates more variation.

Complement Rule

The complement rule is a fundamental concept in probability theory. It states that the probability of an event occurring is equal to 1 minus the probability of it not occurring. This rule is very helpful when you find it easier to calculate the probability of the opposite event. In the exercise, we use the complement rule to find the probability that at least one worker in the sample has a drug problem. First, we calculated the probability that none of the workers has a drug problem, which is denoted as \( P(X = 0) \):\[ P(X = 0) = 0.925^{20} \approx 0.2181 \]The probability that at least one worker has a drug problem is the complement:\[ P(X \geq 1) = 1 - P(X = 0) \approx 1 - 0.2181 = 0.7819 \]This means there is approximately a 78.19% chance that at least one person in the sample of 20 workers will have a drug problem. The complement rule simplifies such calculations and provides a quick way to tackle problems where finding the probability of the original event directly is complex.