Problem 1 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

During the first year at a university that uses a 4 -point grading system, a freshman took ten 3 -credit courses and received two As, three Bs, four Cs, and one \(D\). a. Compute this student's grade-point average. b. Let the random variable \(X\) denote the number of points corresponding to a given letter grade. Find the probability distribution of the random variable \(X\) and compute \(E(X)\), the expected value of \(X\).

Step-by-Step Solution

Answer

a. The student's grade-point average (GPA) is calculated as: GPA = \(\frac{(2 \times 4) + (3 \times 3) + (4 \times 2) + (1 \times 1)}{30}\) b. The probability distribution of the random variable X is: P(X=1) = 1/10 P(X=2) = 4/10 P(X=3) = 3/10 P(X=4) = 2/10 And the expected value E(X) is: E(X) = (1 × 1/10) + (2 × 4/10) + (3 × 3/10) + (4 × 2/10)

Step 1: Convert Letter Grades to Numeric Values and Compute Total Grade Points

To calculate the student's grade-point average, we first need to convert their letter grades to their corresponding numeric values. We are given the following grades: - 2 As - 3 Bs - 4 Cs - 1 D This university uses a 4-point grading system, where A corresponds to 4 points, B to 3 points, C to 2 points, and D to 1 point. We can compute the total grade points by multiplying the number of each letter grade by its corresponding numeric value and summing the products: Total Grade Points = (2 × 4) + (3 × 3) + (4 × 2) + (1 × 1)

Step 2: Compute the Grade-Point Average

Now that we've calculated the total grade points, we can compute the grade-point average (GPA) by dividing the total grade points by the total number of credit hours taken. The student took ten 3-credit courses, for a total of \(10 \times 3 = 30\) credit hours: GPA = Total Grade Points / Total credit hours = \(\frac{(2 × 4) + (3 × 3) + (4 × 2) + (1 × 1)}{30}\)

Step 3: Define the Random Variable X and Find the Probability of Each Value

The random variable X denotes the number of points corresponding to a given letter grade. In this 4-point grading system, X can take on the values 1, 2, 3, or 4, corresponding to D, C, B, or A, respectively. We are given the probabilities of each letter grade for this particular student based on the number of grades received in their first ten courses: - P(X=1) = 1/10 (One D out of ten courses) - P(X=2) = 4/10 (Four Cs out of ten courses) - P(X=3) = 3/10 (Three Bs out of ten courses) - P(X=4) = 2/10 (Two As out of ten courses)

Step 4: Compute the Expected Value of the Random Variable X

With the probabilities of each value of X, we can compute the expected value E(X) using the following formula: E(X) = Σ[x * P(X = x)] In this case, the expected value would be: E(X) = (1 × 1/10) + (2 × 4/10) + (3 × 3/10) + (4 × 2/10) Now we have all the necessary information to answer both questions: a. The student's grade-point average is calculated as: GPA = \(\frac{(2 × 4) + (3 × 3) + (4 × 2) + (1 × 1)}{30}\) b. The probability distribution of the random variable X is: P(X=1) = 1/10 P(X=2) = 4/10 P(X=3) = 3/10 P(X=4) = 2/10 And the expected value E(X) is: E(X) = (1 × 1/10) + (2 × 4/10) + (3 × 3/10) + (4 × 2/10)

Key Concepts

grade-point average

A Grade-Point Average (GPA) is a numerical representation of a student's academic performance, measured over a specific period, like a semester or an academic year. GPA is predominantly calculated in many educational systems to provide a standardized way to assess student achievement.

To calculate GPA, each letter grade is first converted into a numeric value. For instance, in a 4-point grading system, the numeric values assigned are as follows:

• A = 4 points
• B = 3 points
• C = 2 points
• D = 1 point

The student in this example took ten 3-credit courses in total. The formula to compute the GPA is:\[\text{GPA} = \frac{\text{Total Grade Points}}{\text{Total Credit Hours}}\]
Here's how to break it down: Calculate the total grade points by multiplying each grade's numeric value by the number of occurrences of that grade. Then, sum these results to find the total grade points. Finally, divide by the total number of credit hours to get the GPA.

This measure is crucial because it reflects both the quantity of coursework attempted and the level of success in those courses, providing a clear assessment of a student's overall scholastic performance.

expected value

Expected value is a vital concept in probability and statistics used to determine the average outcome you can expect from a random process over the long term. It's especially important when dealing with random variables. To find the expected value, each possible outcome of a random variable is multiplied by its corresponding probability, and then these products are summed up.The mathematical notation for the expected value, often denoted as \(E(X)\), is expressed by the formula:\[E(X) = \sum [x \cdot P(X = x)]\]
In our context, the random variable \(X\) can take values from the possible numeric grades: 1, 2, 3, and 4, each correlated with their respective probabilities. Substituting our values and probabilities into the formula, the expected value becomes:

• \(E(X) = (1 \times \frac{1}{10}) + (2 \times \frac{4}{10}) + (3 \times \frac{3}{10}) + (4 \times \frac{2}{10})\)

This calculation provides the average point value per course the student can expect based on their performance distribution. Expected value helps frame outcomes in the context of average scenarios, ensuring that, given many trials, results tend toward this average.

random variable

A Random Variable is a fundamental concept in probability theory that represents a variable whose possible values are numerical outcomes of a random phenomenon. It provides a mathematical framework for dealing with outcomes that are uncertain.

In our problem, the random variable \(X\) is defined as the number of grade points a student earns, which can be any of these outcomes: 1, 2, 3, or 4. Each value corresponds to a specific letter grade: D, C, B, or A. This translation from actual grades to numerical values allows us to apply statistical measures to analyze student performance.

Understanding the probability distribution of a random variable is critical. This distribution, a listing of all possible values of a random variable and their associated probabilities, provides insight into how data points are spread across different outcomes. In our example, this distribution is:

• \(P(X=1) = \frac{1}{10}\)
• \(P(X=2) = \frac{4}{10}\)
• \(P(X=3) = \frac{3}{10}\)
• \(P(X=4) = \frac{2}{10}\)

By understanding a random variable and its distribution, we gain the ability to calculate various properties, such as the expected value, which gives us a comprehensive view of the likelihoods of different scenarios.