Problem 21 - An Introduction to Mathematical Statistics and Its Applications

Question

Suppose that on-the-job injuries in a textile mill occur at the rate of \(0.1\) per day. (a) What is the probability that two accidents will occur during the next (five-day) workweek? (b) Is the probability that four accidents will occur over the next two workweeks the square of your answer to part (a)? Explain.

Step-by-Step Solution

Answer

The probabilities are calculated using Poisson distribution. It will typically not be the case that the probability of 4 accidents over 2 weeks is the square of the probability of 2 accidents in a week.

Step 1: Setting up the problem

The Poisson distribution formula is given by \(P(x; μ) = (e^{-μ} * μ^{x}) / x!\) where \(μ\) is the average number of events per interval which in our case is 0.1 per day and x is the actual number of successes that result from the experiment. This exercise consists of two parts (a) and (b), Let's first address part (a). For a (five-day) workweek the total rate is \(0.1 * 5 = 0.5\) events per week. The number of accidents we are interested in knowing the probability for is 2.

Step 2: Solve for part (a)

We use the formula to plug in these values: \(P(2; 0.5) = (e^{-0.5} * 0.5^{2}) / 2!\) Solving this will give us the probability of exactly two accidents occurring in a five-day work week.

Step 3: Setting up for part (b)

Now, for part (b), we are asked for the probability of occurrence of 4 accidents over the next two workweeks. Here, the rate is \(0.1 * 5 * 2 = 1\) event per two weeks. And we are interested in 4 events. We are asked to check if this is equal to the square of the answer in part (a), indicating it's not always the case with Poisson distribution.

Step 4: Solve for part (b)

We plug the new values into our Poisson formula: \(P(4; 1) = (e^{-1} * 1^{4}) / 4!\). We solve for this probability and compare whether it's equal to the square of answer in part (a).

Key Concepts

Probability Calculation

Probability calculation can sometimes seem tricky, but the Poisson distribution offers a structured approach to tackle many such challenges. In this scenario, we're using it to model on-the-job injuries at a textile mill that occur at a specific rate. The Poisson formula is expressed as \(P(x; \mu) = (e^{-\mu} * \mu^{x}) / x!\). Here, \(\mu\) represents the average number of events (injuries, in this case) over a given time frame, and \(x\) is the number of occurrences we're interested in. For the first part of the exercise, where we're asked to find the probability of two injuries during a five-day workweek, we calculate \(\mu\) by multiplying the daily rate (0.1) by the number of days (5), giving us a weekly rate of 0.5. We then substitute these values into the formula to find the probability.The Poisson distribution is a versatile tool in various fields because it deals effectively with the discrete nature of event occurrences over a fixed interval.

On-the-Job Injuries

On-the-job injuries are a critical focus in workplace safety and statistical modeling can offer insights that help mitigate risks. Analyzing past data on injury rates, such as keeping track of how often accidents occur, enables companies to predict future occurrences and take preventive measures. In our case, the data suggests that injuries occur at a rate of 0.1 injuries per day. Understanding the average helps in calculating probabilities for different intervals and determining if safety protocols need adjustments. Safety managers can use these statistical outcomes to initiate programs or interventions tailored to reducing injury frequency, as well as train personnel more effectively. Proactively applying these insights not only enhances safety but also optimizes operational efficiency by minimizing downtime related to accidents.

Statistical Modeling

Statistical modeling is a powerful technique in analyzing and understanding complex data sets across various domains. In workplace scenarios like on-the-job injuries, statistical models such as the Poisson distribution can offer valuable predictions. These models help quantify risks and understand the correlation between different variables affecting workplace safety. Through statistical modeling, companies can examine injury patterns over time, predict future trends, and even evaluate the effectiveness of safety interventions. The Poisson distribution is particularly useful when dealing with rare events happening over time or space, making it an ideal choice for modeling injury occurrences. By building accurate statistical models, organizations can not only predict but also prevent potential workplace hazards, thereby ensuring better health outcomes for employees.