Problem 45 - 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. $$\begin{array}{|lc|c|c|c|} \hline \begin{array}{l} \text { Number of Colleges } \\ \text { Applied to } \end{array} & 1 & 2 \text { or } 3 & 4-6 & 7 \text { or more } \\ \hline \begin{array}{l} \text { Percent (as a } \\ \text { decimal) }The student applied to fewer than 4 colleges. \end{array} & 0.20 & 0.29 & 0.37 & 0.14 \end{array}$$$$\begin{aligned} &\text {Using the percents as probabilities, find the probability of}\\\ &\text { each event for a randomly selected student.} \end{aligned}$$

Step-by-Step Solution

Answer

The probability a student applied to fewer than 4 colleges is 0.49.

Step 1: Understand the probabilities

The table provides the percentages in decimal form representing the probabilities for the number of colleges a student applied to. These values as decimal numbers already indicate probabilities.

Step 2: Organize the probabilities

Given probabilities are expressed in decimal forms. The events are: applying to 1 college is 0.20, applying to 2 or 3 colleges is 0.29, applying to 4-6 colleges is 0.37, and applying to 7 or more colleges is 0.14.

Step 3: Probability for less than 4 colleges

To find the probability of a randomly selected student applying to fewer than 4 colleges, we need to add the probabilities of applying to 1 college and 2 or 3 colleges. So, the combined probability is: \(0.20 + 0.29 = 0.49\).

Step 4: Probability for 4 or more colleges

The probability of applying to 4-6 colleges is 0.37, and for 7 or more colleges is 0.14. The probability of applying to 4 or more colleges is obtained by summing these two probabilities: \(0.37 + 0.14 = 0.51\).

Key Concepts

Survey Data Analysis

Survey data analysis involves understanding and interpreting the results from surveys to glean insights. When analyzing survey data, the aim is to translate raw numbers into meaningful statistics that inform decisions.

For instance, in this exercise, the survey results are represented as percentages. Each percentage indicates the proportion of students who applied to a specific number of colleges. In this case, the survey covers how many colleges new students applied to. Breaking down these statistics helps us know where most students fall in terms of their college applications.

Key points to remember when analyzing survey data:

• Always understand what each number represents — in this scenario, percentages of students.
• Consider the total sample size, as larger samples give more reliable information.
• Look for patterns or notable distributions that stand out in the data.

College Application Statistics

Statistics on college applications can provide valuable insights into student behavior and trends. From pressures of applying to multiple colleges to the strategies for higher acceptance rates, these statistics serve multiple purposes.

In our example, the data shows:

• 20% of students applied to just one college.
• 29% applied to 2 or 3 colleges.
• 37% applied to 4-6 colleges.
• 14% applied to 7 or more colleges.

These numbers illustrate a strategy among students to apply to more colleges, possibly to increase their chances of acceptance. Noticeably, the most considerable percentage of students applied to 4-6 colleges.

Such statistics can influence university strategies, like increasing outreach efforts or adjusting admission rates to balance supply with demand.

Probability Distribution

Probability distribution is a mathematical concept that describes the possible outcomes of a random variable and their probabilities. In layman's terms, it shows you what outcomes you can expect and how likely they are.

In our scenario, each probability denotes the likelihood that a random student falls into a specific application bracket. Understanding these distributions helps you grasp how common each behavior is.

• The probability of students applying to fewer than 4 colleges is 0.49, calculated by adding 0.20 (one college) and 0.29 (2 or 3 colleges).
• The probability of students applying to 4 or more colleges is 0.51, which is the sum of 0.37 (4-6 colleges) and 0.14 (7 or more colleges).

These probabilities confirm the general tendency towards applying to multiple colleges. It balances between students opting for fewer applications and those going for more robust application strategies. Recognizing this distribution assists students, advisors, and institutions in making informed choices.