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Chapter 5: Problem 17

According to The Chronicle for Higher Education (Aug. 26, 2011), there were 787,325 Associate degrees awarded by U.S. community colleges in the \(2008-2009\) academic year. A total of 488,142 of these degrees were awarded to women. a. If a person who received an Associate degree in 2008 2009 is selected at random, what is the probability that the selected person will be female? b. What is the probability that the selected person will be male?

Short Answer

Expert verified
The probability that a randomly selected person who received an Associate degree in 2008-2009 is female is \(\frac{488,142}{787,325}\). The probability that this person is male is \(1 - \frac{488,142}{787,325}\).

Step by step solution

01

Identify the Total Number of Degrees Awarded and the Number Awarded to Women

According to the provided data, the total number of Associate degrees awarded by U.S. community colleges in 2008-2009 is 787,325, and the number of degrees awarded to women during this period is 488,142.
02

Calculate the Probability of a Degree Recipient Being Female

The probability of an event can be calculated by dividing the number of successful outcomes by the total number of outcomes. In this case, the successful outcome is a degree awarded to a woman, and the total number of outcomes is the total number of degrees awarded. So the probability \(P (woman)\) is calculated as \(P (woman) = \frac{Number of degrees awarded to women}{Total number of degrees awarded}\) = \(\frac{488,142}{787,325}\).
03

Calculate the Probability of a Degree Recipient Being Male

The probability that a randomly selected degree recipient is male is the complement of the probability that the recipient is female. This can be calculated as \(P(male) = 1 - P(woman)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Statistics Education
Statistics education is a foundational element of modern academic curricula, especially for students pursuing an associate degree. It arms students with the ability to collect, analyze, interpret, and present data in a meaningful way. When applied to real-world scenarios, students learn to make data-driven decisions鈥攁 crucial skill in many fields. Understanding statistics and probability is also essential in interpreting gender distribution throughout various educational tiers, including associate degrees. Engaging with statistics education can thus empower students, not only in their academic journey but also in their future careers.
Probability Theory
Probability theory is a branch of mathematics that deals with quantifying the likelihood of events. This is done by assigning numbers between 0 and 1, where 0 indicates impossibility and 1 denotes certainty. The calculations are based on various models that account for random phenomena. Probability is structured around the concept of events and their outcomes, encapsulating a wide range of applications from weather forecasting to stock market analysis. In the context of the exercise, probability theory was employed to determine the likelihood of selecting a male or female graduate randomly from the pool of associate degree recipients.
Associate Degree
An associate degree is an undergraduate academic program offered by community colleges, vocational schools, and some universities. These programs typically last two years and are designed to provide foundational knowledge and skills in a particular field of study. Students may opt for an associate degree for various reasons, including career advancement, technical skill development, or as a stepping stone toward a bachelor's degree. Understanding the demographics of associate degree recipients, such as gender distribution, can provide insights into educational trends and the workforce landscape.
Gender Distribution in Education
Gender distribution in education refers to the proportionate representation of different genders in educational settings. Over the years, these demographics have shifted significantly across various levels of education. Analyzing data like the number of degrees awarded to men and women, as demonstrated in the exercise, can spotlight trends, inform policy decisions, and improve educational equality. These statistics, when translated into probabilities, can help educators and policymakers understand and address gender disparities in academic achievement and career pathways.

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Most popular questions from this chapter

A large cable company reports that \(80 \%\) of its customers subscribe to its cable TV service, \(42 \%\) subscribe to its Internet service, and \(97 \%\) subscribe to at least one of these two services. (Hint: See Example 5.6\()\) a. Use the given probability information to set up a "hypothetical \(1000 "\) table. b. Use the table from Part (a) to find the following probabilities: i. the probability that a randomly selected customer subscribes to both cable TV and Internet service. ii. the probability that a randomly selected customer subscribes to exactly one of these services.

The Associated Press (San Luis Obispo Telegram-Tribune, August 23,1995 ) reported the results of a study in which schoolchildren were screened for tuberculosis (TB). It was reported that for Santa Clara County, California, the proportion of all tested kindergartners who were found to have TB was 0.0006 . The corresponding proportion for recent immigrants (thought to be a high-risk group) was \(0.0075 .\) Suppose that a Santa Clara County kindergartner is to be selected at random. Are the events selected student is a recent immigrant and selected student has \(T B\) independent or dependent events? Justify your answer using the given information.

A Nielsen survey of teens between the ages of 13 and 17 found that \(83 \%\) use text messaging and \(56 \%\) use picture messaging (鈥淗ow Teens Use Media," Nielsen, June 2009). Use these percentages to explain why the two events \(T=\) event that a randomly selected teen uses text messaging and \(P=\) event that a randomly selected teen uses picture messaging cannot be mutually exclusive.

A rental car company offers two options when a car is rented. A renter can choose to pre-purchase gas or not and can also choose to rent a GPS device or not. Suppose that the events \(A=\) event that gas is pre-purchased \(B=\) event that a GPS is rented are independent with \(P(A)=0.20\) and \(P(B)=0.15\). a. Construct a "hypothetical 1000 " table with columns corresponding to whether or not gas is pre-purchased and rows corresponding to whether or not a GPS is rented. b. Use the table to find \(P(A \cup B)\). Give a long-run relative frequency interpretation of this probability.

Suppose events \(E\) and \(F\) are mutually exclusive with \(P(E)=0.14\) and \(P(F)=0.76\) i. What is the value of \(P(E \cap F) ?\) ii. What is the value of \(P(E \cup F)\) ? b. Suppose that for events \(A\) and \(B, P(A)=0.24, P(B)=0.24\), and \(P(A \cup B)=0.48 .\) Are \(A\) and \(B\) mutually exclusive? How can you tell?
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