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Grades in a Business Course. Indiana University posts the grade distributions for its courses online. 11 Students in Business 100 in the spring 2019 semester received these grades: \(9 \%\) \(\mathrm{A}+, 15 \% \mathrm{~A}, 13 \% \mathrm{~A}-, 10 \% \mathrm{~B}+, 13 \% \mathrm{~B}, 8 \% \mathrm{~B}-7 \% \mathrm{C}+, 11 \% \mathrm{C}, 0 \% \mathrm{C}-, 2 \% \mathrm{D}+, 4 \% \mathrm{D}, 0 \% \mathrm{D}=\), and \(8 \%\) F. Choose a Business 100 student at random. To "choose at random" means to give every student the same chance to be chosen. The student's grade on a four-point scale (with A \(+=4.3\), \(\mathrm{A}=4, \mathrm{~A}-=3.7, \mathrm{~B}+=3.3, \mathrm{~B}=3.0, \mathrm{~B}-=2.7, \mathrm{C}+=2.3, \mathrm{C}=2.0, \mathrm{C}-=1.7, \mathrm{D}+=1.3, \mathrm{D}=1.0\), \(\mathrm{D}=0.7\), and \(\mathrm{F}=0.0\) ) is a random variable \(X\) with this probability distribution: \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|} \hline Value of \(X\) & \(0.0\) & \(0.7\) & \(1.0\) & \(1.3\) & \(1.7\) & \(2.0\) & \(2.3\) & \(2.7\) & \(3.0\) & \(3.3\) & \(3.7\) & \(4.0\) & \(4.3\) \\ \hline Probability & \(0.08\) & \(0.00\) & \(0.04\) & \(0.02\) & \(0.00\) & \(0.11\) & \(0.07\) & \(0.08\) & \(0.13\) & \(0.10\) & \(0.13\) & \(0.15\) & \(0.09\) \\ \hline \end{tabular} a. Is \(X\) a finite or continuous random variable? Explain your answer. b. Say in words what the meaning of \(P(X \geq 3.0)\) is. What is this probability? c. Write the event "the st udent got a grade poorer than \(\mathrm{B}\)-" in terms of values of the random variable \(X\). What is the probability of this event?

Step by step solution

01

Determine the Type of Random Variable

Random variables can either be finite (discrete) or continuous. A finite random variable can take on a countable number of distinct values, whereas a continuous random variable can take on an infinite number of values. Since the student's grade is recorded on a finite set of scale values (like 0.0, 0.7, 1.0, etc.), the random variable \( X \) is finite.

02

Interpret \( P(X \geq 3.0) \)

The expression \( P(X \geq 3.0) \) represents the probability that the student's grade, represented by the random variable \( X \), is at least 3.0. This includes possible grades of \( 3.0, 3.3, 3.7, 4.0, \) and \( 4.3 \).

03

Calculate \( P(X \geq 3.0) \)

To find \( P(X \geq 3.0) \), add the probabilities of \( X \) taking on the values 3.3, 3.7, 4.0, and 4.3.\[ P(X = 3.0) = 0.13, \]\[ P(X = 3.3) = 0.10, \]\[ P(X = 3.7) = 0.13, \]\[ P(X = 4.0) = 0.15, \]\[ P(X = 4.3) = 0.09 \]Thus, \[ P(X \geq 3.0) = 0.13 + 0.10 + 0.13 + 0.15 + 0.09 = 0.60 \].

04

Define the Event of a Grade Poorer than B-

Grades poorer than B- include C+, C, D+, D, and F. These correspond to \( X \) values of 0.0, 1.0, 1.3, 2.0, and 2.3. Since B- corresponds to 2.7, the grades poorer than B- are less than 2.7.

05

Calculate the Probability of Grades Less than B-

To determine \( P(X < 2.7) \), sum the probabilities of the values of \( X \) that are less than 2.7:\[ P(X = 0.0) = 0.08, \]\[ P(X = 0.7) = 0.00, \]\[ P(X = 1.0) = 0.04, \]\[ P(X = 1.3) = 0.02, \]\[ P(X = 1.7) = 0.00, \]\[ P(X = 2.0) = 0.11, \]\[ P(X = 2.3) = 0.07 \]Summing these gives:\[ P(X < 2.7) = 0.08 + 0.00 + 0.04 + 0.02 + 0.00 + 0.11 + 0.07 = 0.32 \].

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Key Concepts

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

Probability Distributions

Probability distributions are essential in understanding how likely different outcomes are in a random scenario. When talking about grades, a probability distribution will show the likelihood of each grade occurring. For instance, in a business course, students might receive grades ranging from A+ to F. The probability distribution will assign a probability to each possible grade, reflecting each grade's frequency or possibility among students.
Such distributions can be visualized in different ways, commonly using tables or graphs. In our case, a table summarizes the probability that a student receives a specific grade, showing grades and their corresponding probabilities.
Distributions are not only useful for interpreting current data but also predicting potential outcomes. This makes them fundamental for dealing with random situations, providing insight into both individual likelihoods and conceivable patterns.

Discrete Variables

Discrete variables are a type of random variable that can take on a countable number of distinct values. In the context of grade distributions, the grades themselves act as discrete variables. The grades are distinct and separate, like A, B+, C, etc. The values assigned to these grades on a 4-point scale, such as 0.0, 0.7, and 4.3, are also discrete since they form a finite list of possible outcomes.
Discrete variables contrast with continuous variables, which can assume an infinite number of values within a given range. While continuous variables flow smoothly from one value to another, discrete variables "jump" from one possible value to the next. When analyzing situations involving discrete variables, it is often simpler to calculate the probability of specific outcomes.

• They are countable, making them easier to work with mathematically.
• They form categories, such as grades or test scores.

Grade Distributions

Grade distributions show the spread of students' grades across different categories, from top-performing to underperforming. It's essentially a way to understand students' achievements in a course by looking at how many students fall into each grade category.
By examining the distribution, educators can spot trends, such as how many students achieved an "A" compared to those who received lower grades. This can help identify areas where students might need more support or where teaching methods could be improved.

• An even distribution shows balanced student performance.
• A skewed distribution may indicate teaching areas needing revision.
• Grade distributions are crucial for academic assessment and planning.

Understanding grade distributions also helps students see where they stand and what is common performance-wise among their peers.

Probability Calculations

Probability calculations involve determining the likelihood of specific outcomes based on the given distribution data. In our example, you might calculate the probability that a randomly chosen student scored above a certain grade or below. This is done by summing up the probabilities of all possible grades that meet specific criteria.
For example, to calculate the probability of a grade less than B-, you would sum the probabilities for C+, C, D+, D, and F, each correlating with specific numerical values like 0.0 and 1.0.
Calculating probabilities allows us to make predictions and informed decisions. We consider all available possibilities and weight them by how likely each scenario is.

• It helps in making predictions about student performance.
• Allows for strategic planning in addressing student needs.
• Can guide adjustments to teaching methods.

In any field, probability calculations are powerful tools for understanding and planning around uncertainty.