In Exercises 13 and 14, use the table, which shows the approximate distribution of the sizes of firms for a recent year. (Adapted from North American Industry Classification System)

13. Find the probability that a randomly selected firm will have more than four employees.

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with finding the probability that a randomly selected firm will have more than four employees. This means you need to consider all firms with 5 or more employees.
2. Step 2: Identify the relevant data from the table. The table provides the percentage of firms in different employee size categories: '1 to 4', '5 to 9', '10 to 19', '20 to 99', and '100 or more'.
3. Step 3: Focus on the categories that represent firms with more than four employees. These categories are '5 to 9', '10 to 19', '20 to 99', and '100 or more'. The corresponding percentages are 11.5%, 4.9%, 4.0%, and 1.1%.
4. Step 4: Add the percentages for these categories to find the total probability. Use the formula: \( P(>4 \, \text{employees}) = P(5 \text{ to } 9) + P(10 \text{ to } 19) + P(20 \text{ to } 99) + P(100 \text{ or more}) \).
5. Step 5: Convert the total percentage to a probability by dividing by 100. This will give the probability that a randomly selected firm has more than four employees.

Key Concepts

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, it refers to the chance of randomly selecting a firm with more than four employees from the given distribution. Understanding how to calculate probability is essential for interpreting statistical data and making informed decisions based on that data.

Cumulative Distribution

A cumulative distribution function (CDF) represents the probability that a random variable takes on a value less than or equal to a certain number. To find the probability of selecting a firm with more than four employees, one must calculate the cumulative percentage of firms with four or fewer employees and subtract it from 100%. This concept is crucial for understanding how probabilities accumulate across different categories.

Percentages in Statistics

Percentages are a way to express a number as a fraction of 100, making it easier to compare different data sets. In the context of the table, the percentages represent the proportion of firms within specific employee ranges. Understanding how to interpret and manipulate these percentages is vital for answering questions about distributions and probabilities in statistics.