Question

You are conducting a survey on the use of air-plane phones. You survey 320 adults and find that 288 of them never made a phone call from an airplane. If you surveyed 3500 adults, how many of them would you predict have made a phone call from an airplane? Explain.

Short answer

The predicted number of adults out of 3500 who have made a phone call from an airplane is 350.

Step-by-step solution

Find the proportion of adults who never made a phone call from an airplane

In the survey of 320 adults, 288 have never made a phone call from an airplane. Subtract 288 from 320 to find the number of adults who have made a phone call from an airplane. The calculation is: 320 - 288 = 32. So, the proportion of adults who have made a phone call from an airplane is \(\frac{32}{320}\)

Set up and solve for the unknown

The problem assumes that the proportion of adults who have made a phone call from an airplane will be the same in a larger survey of 3500 adults. Set up the equation as:\(\frac{x}{3500} = \frac{32}{320}\), where x represents the number of adults out of 3500 who have made a phone call from an airplane. To solve for x, cross multiply and solve for x. The calculation is: \(x = \frac{32}{320} * 3500\)

Calculate and interpret the result

Perform the multiplication to find value of x. Then round to the nearest whole number if necessary because the number of adults who have made a phone call from an airplane should be an integer. The number of adults out of 3500 who have made a phone call from an airplane, based on the given proportion, is the value of x. Interpret the result in the context of the problem.

Key concepts

Ratios and Proportional Relationships

Understanding ratios and proportional relationships is fundamental in solving real-world problems, such as predicting behaviors or outcomes in larger populations based on a sample. In the given exercise, we establish a ratio to represent a relationship between two quantities: the number of adults who have made a phone call from an airplane and the total number surveyed.

Proportions are equations that express two ratios are equivalent. For instance, if we have the ratio of adults who have called from an airplane to the total surveyed (32 out of 320), and we want to predict the outcome for a different total number of adults (3500), we use a proportion to maintain this relationship. The key is to understand that if the ratio remains constant, and we adjust one part of the proportion, the other must change in a way that this constant relationship is preserved.

Predicting Outcomes

The concept of predicting outcomes is particularly useful when making informed guesses about larger datasets based on a smaller sample. The critical assumption here is that the sample is representative of the larger group. Given the proportion of adults from the survey who have not made calls from an airplane, we can predict how many out of a larger group could potentially have made those calls.

The exercise involves scaling up the original ratio of those who made calls to apply it to a larger number of people. It's the basis for making predictions in various fields like statistics, economics, and social sciences, allowing us to analyze trends and make forecasts without needing to collect data for the entire population. This method saves time and resources while providing valuable insights.

Cross Multiplication

When we encounter a problem that involves finding an unknown variable in a proportion, cross multiplication becomes a useful tool. The technique involves multiplying the numerator of one ratio by the denominator of the other ratio and setting the results equal to each other to solve for the unknown. In our exercise, the equation \[\begin{equation}\frac{x}{3500} = \frac{32}{320}\end{equation}\] can be solved by cross multiplying to get \[\begin{equation}320x = 32 \times 3500\end{equation}\].Then, dividing both sides by 320 will isolate the variable x, providing us with the number of adults we predict will have made a phone call from an airplane. This step is crucial for converting our understanding of ratios into a numeric prediction.

Problem-solving Strategies

Effective problem-solving strategies are at the core of mathematics and help us navigate a wide array of challenges. To systematically approach a problem, first, understand what is being asked. In our case, it’s the prediction of behavior in a larger population based on a sample. Next, identify the information given and organize it: the ratio of adults who have made a phone call from an airplane to the total surveyed.

Then, determine the process necessary to find the solution: using ratios, set up a proportional relationship and solve using cross multiplication. Finally, perform any needed calculations and interpret the result in context. This structured diligence ensures clarity of thought and method, which is essential for accurate and reliable solutions in mathematical modeling and beyond.