Problem 30 - Basic Statistics for Business and Economics with Student

Question

Textbook authors and publishers work very hard to minimize the number of errors in a text. However, some errors are unavoidable. Mr. J. A. Carmen, statistics editor, reports that the mean number of errors per chapter is 0.8 . What is the probability that there are less than 2 errors in a particular chapter?

Step-by-Step Solution

Answer

The probability is approximately 0.809.

Step 1: Identify the Probability Distribution

The scenario describes the number of errors per chapter, which follows a Poisson distribution because it counts the number of events (errors) in a fixed interval (a chapter). The exercise provides the mean number of errors per chapter as 0.8, so the parameter \( \lambda \) for the Poisson distribution is 0.8.

Step 2: Formula for Poisson Probability

The probability of a Poisson random variable \( X \) with parameter \( \lambda \) taking on the value \( k \) is given by: \[ P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \]Where \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

Step 3: Compute Probabilities for k=0 and k=1

Calculate the probabilities for fewer than 2 errors, which means \( k=0 \) and \( k=1 \).For \( k = 0 \):\[ P(X = 0) = \frac{e^{-0.8} \cdot 0.8^0}{0!} = e^{-0.8} \]For \( k = 1 \):\[ P(X = 1) = \frac{e^{-0.8} \cdot 0.8^1}{1!} = 0.8 \cdot e^{-0.8} \]

Step 4: Calculate Total Probability

Find the probability of having less than 2 errors by summing the probabilities calculated:\[ P(X < 2) = P(X = 0) + P(X = 1) \]Where:\[ e^{-0.8} \] is calculated to be approximately 0.44933.So,\[ P(X < 2) = 0.44933 + 0.8 \times 0.44933 = 0.44933 + 0.35946 = 0.80879 \]

Step 5: Conclusion

Thus, the probability that there are less than 2 errors in a particular chapter is approximately 0.809.

Key Concepts

probability distribution

In statistics, a probability distribution provides a mathematical function that describes the likelihood of different outcomes in a random phenomenon. For example, when considering the number of errors found in a textbook chapter, we can use a probability distribution to predict the frequency of various occurrences. This problem, specifically, involves the Poisson distribution.

The Poisson distribution is often used when we are counting the number of times an event occurs within a fixed interval, which could be time, area, or any other measurable quantity. It is particularly useful for rare events. In our context, it represents the likelihood of a specific number of errors within a chapter.

• The variable of interest is discrete, i.e., the number of errors.
• We assume a constant mean rate of occurrence, denoted by the parameter \(\lambda\).
• Events are independent, meaning the occurrence of one event does not affect the others.

By setting these assumptions, the Poisson distribution helps model the uncertainty and randomness in such scenarios effectively.

mean number of errors

The mean number of errors, often denoted by \(\lambda\), is a crucial part of understanding how a Poisson distribution functions. In this problem, the mean number of errors per chapter is given as 0.8.

This figure, \(\lambda = 0.8\), represents not only the average number of errors per single chapter but also serves as the core parameter for the Poisson probability formula. When dealing with Poisson distributions, the mean is equal to the variance, which highlights the unique property of this distribution.

• \(\lambda\) is always positive as it counts the average occurrence of events.
• It sets the expectation of error occurrences within any given chapter.
• Establishes the typical rate at which errors could happen and helps in predicting various probabilities.

Understanding the mean and its role facilitates better comprehension of the probability calculations that follow in the problem.

probability calculation

Probability calculation involves determining the likelihood of specific outcomes. Within the Poisson distribution framework, this relies on a particular formula to quantify the chances of different events.

The probability of finding a certain number, \(k\), of errors in a chapter is expressed with the formula:
\[ P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \]
where \(e\) is approximately 2.71828, representing the base of natural logarithms.

• Given \(\lambda = 0.8\), different probabilities are calculated by substituting \(k\) with the potential number of errors that might occur (e.g., \(k = 0\) or \(k = 1\)).
• The formula considers both the rate parameter and factorial of \(k\), showing how rare larger numbers of errors are.
• By computing probabilities for smaller \(k\), and summing where necessary, we determine the overall likelihood of specific outcomes.

This methodical approach highlights the utility of the Poisson distribution in precise probability estimations.

fixed interval events

Fixed interval events refer to occurrences measured over a set period or space. In this context, we count errors in a single textbook chapter, considered a fixed interval due to its defined boundaries. This is a common application of the Poisson distribution: assessing frequent random events over predefined domains.

The essence of fixed interval analysis is that while the timing or exact positioning of an event can vary, the interval within which it is observed remains constant. Some key characteristics include:

• Events are assumed to happen independently within the interval, each event distinct and unaffected by previous occurrences.
• The number of possible outcomes within the interval generally follows the Poisson distribution.
• Boundaries of the interval determine constraints and expectations as per statistical modeling.

By focusing on fixed interval events, particularly in realms like error-checking, statistical tools like Poisson distributions provide valuable insights into future probabilities and patterns.