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Chapter 5: Problem 56

Mac or PC? A recent census at a major university revealed that 40% of its students primarily used Macintosh computers (Macs). The rest mainly used PCs. At the time of the census, 67% of the school鈥檚 students were undergraduates. The rest were graduate students. In the census, 23% of respondents were graduate students who said that they used PCs as their main computers. Suppose we select a student at random from among those who were part of the census. (a) Assuming that there were 10,000 students in the census, make a two-way table for this chance process. (b) Construct a Venn diagram to represent this setting. (c) Consider the event that the randomly selected student is a graduate student who uses a Mac. Write this event in symbolic form using the two events of interest that you chose in (b). (d) Find the probability of the event described in (c). Explain your method.

Short Answer

Expert verified
The probability that a randomly selected student is a graduate who uses a Mac is 0.1.

Step by step solution

01

Understand and Set Up the Problem

We're dealing with a categorical data problem where we need to analyze different groups of university students based on their choice of computer and their educational status. We're given percentages for various combinations and need to analyze a sample of 10,000 students.
02

Construct a Two-Way Table

To organize the data, we construct a two-way table with the categories: 'Undergraduate' and 'Graduate' for rows, and 'Mac User' and 'PC User' for columns. Begin filling out the table using given percentages: - Total Students: 10,000 - Mac Users: 40% of 10,000 = 4,000 - PC Users: 60% of 10,000 = 6,000 - Undergraduate Students: 67% of 10,000 = 6,700 - Graduate Students: 33% of 10,000 = 3,300 - Graduate students using PC: 23% of 10,000 = 2,300
03

Fill in the Missing Table Values

Determine other missing values using subtraction: - Graduate Mac Users = Total Graduates - Graduate PC Users = 3,300 - 2,300 = 1,000 - Undergraduate PC Users = Total PC Users - Graduate PC Users = 6,000 - 2,300 = 3,700 - Undergraduate Mac Users = Total Students - (Undergrad PC + Grad PC + Grad Mac) = 10,000 - (3,700 + 2,300 + 1,000) = 3,000 Now the table looks like: | | Mac | PC | |---------|-----|-------| | Undergr | 3000| 3700 | | Grad | 1000| 2300 | | Total | 4000| 6000 |
04

Construct the Venn Diagram

Using the data from the table, we create two overlapping circles to represent Mac and Graduate users. Calculate the intersections: - Intersection (Graduate and Mac Users) = 1,000, - Only Mac Users = Total Mac Users - Graduate Mac Users = 4,000 - 1,000 = 3,000 - Only Graduate Users = Total Graduates - Graduate Mac Users = 3,300 - 1,000 = 2,300 (PC Users accounted separately). The non-overlapping region of PCs would then be Undergraduate PCs (3,700).
05

Symbolic Representation of the Event

Define event A as 'Graduate student' and event B as 'Uses a Mac'. The event we're interested in (graduate student using Mac) can be represented as \( A \cap B \). This is the intersection of Graduate students and Mac users.
06

Calculate Probability of the Event

The probability of a graduate student using a Mac, which is event \( A \cap B \), is calculated by dividing the number of Graduate students using Macs by the total number of students:\[ P(A \cap B) = \frac{1000}{10000} = 0.1 \]Therefore, the probability is 0.1.

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Key Concepts

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

Venn Diagram
A Venn Diagram is a visual tool that helps us understand the relationship between different sets or groups.
In this exercise, we are analyzing students based on their computer usage and educational status. We use a Venn Diagram to represent two important groups: student Mac users and graduate students.
This allows us to visualize how these groups overlap. The Venn Diagram circles intersect, showing students who are both graduate students and Mac users.
This overlap represents the event that we want to analyze further. Each part of the Venn Diagram provides a clear depiction of students who use only Macs, only PCs, and also those who belong to both groups.
If you're dealing with complex categorical data, Venn Diagrams help you see how groups share common elements or are distinct from each other. This visual aid is crucial for getting a clear picture of the data presented.
Probability Calculation
Probability Calculation involves finding how likely an event is to occur based on known data. In this example, we want to find the probability that a randomly selected student is a graduate student using a Mac.
To calculate this, we first identify the event of interest, which is the intersection of two events: being a graduate student and using a Mac.
The formula for probability is straightforward: divide the number of favorable outcomes by the total number of possible outcomes.
Here, the favorable outcomes are the 1,000 graduate students who use Macs, and the total outcomes are all students surveyed, which is 10,000.
This gives us the probability as \[ P(A \cap B) = \frac{1000}{10000} = 0.1 \]This simple division tells us that there's a 10% chance for a randomly picked student to be a graduate Mac user. Probability calculations like this one allow us to make informed guesses about groups within a population and see the broader picture through numerical representation.
Categorical Data Analysis
Categorical Data Analysis is a method for examining data that can be divided into categories, helping us to make sense of different groups within a dataset.
In this scenario, we categorize students by their type of computer and educational level.
This approach is vital as it simplifies complex data sets by organizing them into understandable categories.
By constructing a two-way table, we match our two categories鈥攃omputer type (Mac or PC) and education level (Undergraduate, Graduate)鈥攁llowing us to count the number of students in each subgroup.
This analysis reveals relationships and patterns, such as how many students are graduate Mac users versus undergraduate PC users.
Through Categorical Data Analysis, we can easily spot trends, compare groups, and make informed predictions.
  • Two-way tables provide a neat, clear view of data.
  • They help decode large amounts of information.
  • They are essential for further probability calculations and data visualizations like Venn Diagrams.
When dealing with any categorical data, this kind of analysis is foundational for making logical data-driven decisions.

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Most popular questions from this chapter

Bright lights? A string of Christmas lights contains 20 lights. The lights are wired in series, so that if any light fails the whole string will go dark. Each light has probability 0.02 of failing during a 3-year period. The lights fail independently of each other. Find the probability that the string of lights will remain bright for 3 years.

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