91Ó°ÊÓ

A population distribution provides a snapshot of how different categories or values are represented across a population. In our example, the sizes of pizzas fall into four categories—12 inches, 14 inches, 16 inches, and 18 inches. The given proportions for these categories reflect the distribution of single-pizza orders. Over a long sequence of orders, one can expect:

• 20% of pizzas to be 12 inches.
• 25% to be 14 inches.
• 50% to be 16 inches.
• Only 5% to be 18 inches.

This distribution helps reveal patterns or tendencies, such as how the 16-inch pizza is the most popular choice, and the 18-inch pizza is the least favored. Understanding the population distribution allows businesses to make informed decisions, like adjusting inventory or marketing strategies.

In statistics, proportions represent the size or share of each category related to the total. For pizza shop data, proportions describe the likelihood of ordering each pizza size given previous orders. The proportions for sizes of 12, 14, 16, and 18 inches are 0.20, 0.25, 0.50, and 0.05, respectively.
Understanding proportions is key in making predictions. For example, given the proportion of 0.50, you could say that if 100 pizzas are ordered, around 50 of them might be 16-inch pizzas. Proportions help quantify the preferences within a population, assisting in inventory planning, customer satisfaction analysis, and more. These proportions should be monitored regularly as they may evolve over time due to changes in consumer preferences or menu updates.

Calculating probabilities from proportions is straightforward. It involves summing the proportions of interest. For instance, the probability of ordering a pizza less than 16 inches, denoted as \(P(x < 16)\), can be found by adding the probabilities of ordering 12-inch and 14-inch pizzas:

• \(P(x < 16) = P(x = 12) + P(x = 14) = 0.20 + 0.25 = 0.45\)

Thus, there's a 45% chance that a pizza ordered will be less than 16 inches. Similarly, to find \(P(x \leq 16)\), which includes ordering a 16-inch pizza, sum up:

• \(P(x \leq 16) = P(x = 12) + P(x = 14) + P(x = 16) = 0.20 + 0.25 + 0.50 = 0.95\)

This means there's a 95% chance of ordering a pizza that is 16 inches or smaller. Such calculations are essential for anticipating demand and aligning with customer preferences.

Descriptive statistics afford a method to summarize and describe a set of data. With pizza orders, key statistics like mean, median, and mode are useful to summarize pizza size preferences. For our dataset, the mode is apparent as 16-inch since it has the highest proportion.

Descriptive statistics simplify how we interpret and communicate data patterns. Outside mean, median, and mode, proportions themselves serve as a form of descriptive statistics, reflecting frequency and potential variances in orders. Descriptive approaches inform better strategic decisions within the business, like what deals to promote or when certain stock may run low. They play an essential role in both understanding historical data and forecasting future trends.