Problem 46 - 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 at least 2 colleges.

Step-by-Step Solution

Answer

The probability is 0.80.

Step 1: Identify the Probability Categories

From the given table, identify the probability categories relevant to our question. We're looking for the category 'at least 2 colleges', which includes '2 or 3', '4-6', and '7 or more' colleges.

Step 2: Extract Relevant Probabilities

From the table, the probabilities as decimals for '2 or 3', '4-6', and '7 or more' colleges are 0.29, 0.37, and 0.14 respectively.

Step 3: Sum the Probabilities

To find the probability that a student applied to at least 2 colleges, add the probabilities of applying to '2 or 3', '4-6', and '7 or more' colleges: \[0.29 + 0.37 + 0.14\]

Step 4: Calculate the Total Probability

Perform the addition: \[0.29 + 0.37 + 0.14 = 0.80\]This represents the probability that a randomly selected student applied to at least 2 colleges.

Key Concepts

Survey Analysis

Survey analysis involves collecting data via surveys and examining the responses to gain insights or draw conclusions. Surveys are a common method used in research to gather quantitative data. In this particular exercise, a survey was conducted among 282,549 freshmen to understand their application behavior to colleges.

Key aspects of survey analysis include:

• Data Collection: Gathering responses from individuals, which are subsequently compiled into a dataset. This is the first step in understanding participant behavior or opinions.
• Data Interpretation: Interpreting survey results involves converting raw numbers into meaningful information. In our scenario, the results summarized the application patterns of students to baccalaureate colleges.
• Probability Analysis: One way to interpret the survey data is through probability. By converting percentages into probabilities, predictions about future behavior can be made, as shown in the student's probability of applying to a certain number of colleges.

Understanding and analyzing survey data helps in decision-making processes and in drawing conclusions about large groups, or populations, based on a smaller sample.

Baccalaureate Colleges

Baccalaureate colleges are institutions primarily focused on undergraduate education with a major emphasis on awarding bachelor's degrees. In this exercise, these colleges play a crucial role in the survey, as it captures freshmen's behavior towards applying to such institutions.

Here are some essential points about baccalaureate colleges:

• Focus on Undergraduate Education: They primarily offer undergraduate programs, providing a solid foundation in a wide array of disciplines.
• Diverse Enrollment: These institutions cater to a diverse student body. A survey of baccalaureate colleges can provide comprehensive insights into student choices.
• Importance in Higher Education: They play a vital role in the educational landscape by providing accessible higher education opportunities across the country.

The results of the survey among freshmen applying to baccalaureate colleges offer valuable insights into college application trends, highlighting the pivotal role these institutions play in the higher education sector.

Random Selection

Random selection is a method used to ensure that every individual or element has an equal opportunity to be chosen. It is crucial in survey research to guarantee unbiased results.

When analyzing the survey data from freshmen applications:

• Ensures Representation: Random selection helps in making sure that the sample is representative of the entire population, which would include all potential freshman applicants.
• Reduces Bias: By randomly selecting students, any particular characteristic doesn't skew the results. This allows for a more generalizable conclusion.
• Facilitates Probability Analysis: Knowing that the selection is random adds validity to probability calculations. If data is collected randomly, conclusions drawn about probabilities are more likely to reflect the true scenario.

Using random selection within survey studies is fundamental in creating reliable datasets that accurately reflect the population's behaviors and opinions.