Question

A pizza shop sells pizzas in four different sizes. The 1,000 most recent orders for a single pizza resulted in the following proportions for the various sizes: $$ \begin{array}{lcccc} \text { Size } & 12 \text { in. } & 14 \text { in. } & 16 \text { in. } & 18 \text { in. } \\ \text { Proportion } & 0.20 & 0.25 & 0.50 & 0.05 \end{array} $$ With \(x=\) the size of a pizza in a single-pizza order, the given table is an approximation to the population distribution of \(x\). a. Write a few sentences describing what you would expect to see for pizza sizes over a long sequence of single-pizza orders. b. What is the approximate value of \(P(x<16)\) ? c. What is the approximate value of \(P(x \leq 16) ?\)

Short answer

1. In a long sequence of pizza orders, the most ordered pizzas would be the 16-inch pizzas, followed by the 14-inch pizzas, and the least orders would be for the 18-inch pizzas. 2. The approximate value of \(P(x<16)\) is \(0.45\). 3. The approximate value of \(P(x \leq 16)\) is \(0.95\).

Step-by-step solution

Interpretation of the distribution

Interpret the provided distribution. The distribution can be read as such: 20% of the orders are for 12-inch pizzas, 25% are for 14-inch pizzas, 50% are for 16-inch pizzas and 5% are for 18-inch pizzas. Therefore, in a long sequence of orders, mostly 16-inch pizzas should be seen, followed by 14-inch pizzas. The smallest number of orders would be for the largest pizza size, the 18-inch pizza.

Calculate \(P(x

To calculate the probability \(P(x<16)\), sum the proportions of pizzas with sizes less than 16 inches, that are the 12-inch and 14-inch pizzas. Therefore, the probability is \(P(x<16) = 0.20 + 0.25 = 0.45\).

Calculate \(P(x \leq 16)\)

The probability that the size of a pizza is less than or equal to 16, \(P(x \leq 16)\), considers additionally the 16-inch pizzas. Thus, this probability is the sum of probabilities of the 12-inch, 14-inch, and 16-inch pizzas. Hence, it is calculated as follows, \(P(x \leq 16) = 0.20 + 0.25 + 0.50 = 0.95\).

Key concepts

Population Distribution

When we mention population distribution, we're referring to the way different sizes of pizzas are ordered within a large group or 'population' of orders.

In the context of the pizza shop example, the population distribution provides us a picture of customers' preferences for pizza sizes. It quantifies the proportion of each size ordered out of a thousand recent single-pizza orders. If we imagine extending this pattern across many more orders, we can predict that the distribution would remain relatively stable. The distribution shows that most customers prefer a 16-inch pizza, while a 12-inch pizza is the least preferred.

Understanding this distribution is crucial for the pizza shop as it helps in planning inventory and ensuring that customer preferences are met most efficiently. This real-world reflection of customer choice is foundational in probability and statistics, helping us to make predictions and informed decisions based on data.

Probabilistic Modeling

The practice of using statistical methods to forecast future events or behaviors is known as probabilistic modeling. We are attempting to describe real-world phenomena where the outcome is uncertain, like the size of pizza that will be ordered next.

Probabilistic models are built using observed data - in this case, the proportion of different pizza sizes ordered out of 1,000 orders. With this model, we can predict the likelihood of future pizza orders. This form of modeling is widely used in various fields, from finance to meteorology, where predicting future events with some level of certainty can have significant implications.

Using the pizza shop data, the probabilistic model allows us to make informed estimates about what proportion of each pizza size will be ordered in the next sequence of sales, and guides decisions such as how many of each size to prepare ahead of a busy period.

P(x<16)

The probability notation P(x<16) quantifies the likelihood that a single-pizza order will be for a pizza that is less than 16 inches in diameter.

From the pizza shop's data, we can determine this by adding up the proportions of the two smaller sizes available, since those are the sizes smaller than 16 inches. Hence, we calculate it as \(P(x<16) = 0.20 + 0.25 = 0.45\). This means there's a 45% chance that any given single-pizza order will be for a size less than 16 inches, which helps the shop anticipate demand more accurately.

P(x<=16)

Now, the notation P(x<=16) expresses the probability that a single-pizza order is for a pizza that is 16 inches or smaller.

To find this probability, we need to add the proportions of all pizza sizes up to and including the 16-inch pizzas. Following the data from the problem, we calculate this as \(P(x \leq 16) = 0.20 + 0.25 + 0.50 = 0.95\). This indicates that 95% of the orders are expected to be for pizzas that are 16 inches or smaller, which can inform the pizza shop’s preparation and inventory management to ensure customer satisfaction.