Problem 47 - A Graphical Approach to College Algebra

Question

The table gives the results of a survey of \(282,549\) freshmen from a recent class year at 437 of the nation's baccalaureate colleges and universities. $$\begin{array}{|lc|c|c|c|} \hline \begin{array}{l} \text { Number of Colleges } \\ \text { Applied to } \end{array} & 1 & 2 \text { or } 3 & 4-6 & 7 \text { or more } \\ \hline \begin{array}{l} \text { Percent (as a } \\ \text { decimal) }The student applied to fewer than 4 colleges. \end{array} & 0.20 & 0.29 & 0.37 & 0.14 \end{array}$$$$\begin{aligned} &\text {Using the percents as probabilities, find the probability of}\\\ &\text { each event for a randomly selected student.} \end{aligned}$$The student applied to more than 3 colleges.

Step-by-Step Solution

Answer

The probability is 0.51.

Step 1: Understand the Table Data

The table provides the distribution of application numbers for 282,549 freshmen across four categories: 1 college (0.20), 2 or 3 colleges (0.29), 4 to 6 colleges (0.37), and 7 or more colleges (0.14). The percentage (as a decimal) next to each range describes the likelihood of students applying to that number of colleges.

Step 2: Define the Event of Interest

We need to find the probability that a randomly chosen student applied to more than 3 colleges. This corresponds to the categories "4-6" and "7 or more." Let's refer back to the table to verify these percentages.

Step 3: Extract Relevant Probabilities

From the table, the probability that a student applied to 4 to 6 colleges is 0.37, and the probability that a student applied to 7 or more colleges is 0.14.

Step 4: Calculate the Probability for More Than 3 Colleges

To find the probability that a student applied to more than 3 colleges, add the probabilities of the relevant events: \[P(4-6 \, ext{colleges}) + P(7 \, ext{or more colleges}) = 0.37 + 0.14\]

Step 5: Compute the Result

Perform the addition to find the total probability: \[0.37 + 0.14 = 0.51\]Therefore, the probability that a student applied to more than 3 colleges is 0.51.

Key Concepts

Survey Data Analysis

Survey data analysis is a powerful tool for interpreting data collected from surveys. It involves understanding and analyzing data to draw meaningful conclusions. In this context, we analyzed survey data from freshmen about the number of colleges they applied to.
The survey provides information on distinct categories like '1 college', '2 or 3 colleges', '4-6 colleges', and '7 or more colleges.' For each category, the percentage assigned, such as 0.20 for 1 college, is extracted and used to gauge probabilities.

• Survey data offers an empirical view of behaviors or preferences.
• Analyzing survey data can help in decision-making or strategizing.
• Probabilities derived from survey data represent real-world trends.

Understanding survey data helps bridge the gap between raw numbers and actionable insights. It transforms numbers into narratives reflecting student application behaviors.

Baccalaureate Colleges

Baccalaureate colleges are institutions that primarily focus on undergraduate education, granting mostly bachelor's degrees. They include a wide range of disciplines and are different from universities that also offer advanced post-graduate degrees.
This survey involved 437 baccalaureate colleges and informed us of the application patterns of their students. Knowing the type of institution in such analyses provides context, allowing for more targeted insights.

• Location, size, and focus of baccalaureate colleges can influence application trends.
• A diverse range of colleges facilitates comprehensive survey results.
• Student preferences and trends can be better understood through this dataset.

By focusing on these institutions, the survey gives a clear picture of how application patterns might be influenced by the unique offerings of baccalaureate colleges.

Decimal Probabilities

Decimal probabilities are probabilities expressed in decimal form rather than a fraction or percentage. They are crucial in survey data analysis as they offer a more standardized way to calculate the likelihood of various events.
In the exercise, probabilities like 0.37 for applying to '4 to 6 colleges' or 0.51 for 'more than 3 colleges' were derived. These decimals simplify calculating compound probabilities by allowing straightforward addition or multiplication.

• Decimal form is intuitive for calculations and comparisons.
• Probabilities range between 0 and 1, where 0 means an impossible event and 1 indicates certainty.
• Decimals are flexible and help in dealing with large datasets efficiently.

By employing decimal probabilities, it's easier to understand and communicate the chances of different outcomes, making complex data more accessible.