Problem 9 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The grade distribution for a certain class is shown in the following table. Find the probability distribution associated with these data. $$ \begin{array}{lccccc} \hline \text { Grade } & \text { A } & \text { B } & \text { C } & \text { D } & \text { F } \\ \hline \text { Frequency of } & & & & & \\ \text { Occurrence } & 4 & 10 & 18 & 6 & 2 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The probability distribution for the given grade distribution is: $$ \begin{array}{c|c} \text{Grade} & \text{Probability} \\ \hline A & \frac{1}{10} \\ B & \frac{1}{4} \\ C & \frac{9}{20} \\ D & \frac{3}{20} \\ F & \frac{1}{20} \\ \end{array} $$

Step 1: Calculate the total number of students

Sum the frequencies for all grades: Total students = 4 (A) + 10 (B) + 18 (C) + 6 (D) + 2 (F) = 40 students Now that we know the total number of students in the class, we can calculate the probability distribution by dividing the frequency of each grade by the total number of students.

Step 2: Calculate the probability of grade A

Divide the frequency of grade A by the total number of students: P(A) = \(\frac{4}{40} = \frac{1}{10}\)

Step 3: Calculate the probability of grade B

Divide the frequency of grade B by the total number of students: P(B) = \(\frac{10}{40} = \frac{1}{4}\)

Step 4: Calculate the probability of grade C

Divide the frequency of grade C by the total number of students: P(C) = \(\frac{18}{40} = \frac{9}{20}\)

Step 5: Calculate the probability of grade D

Divide the frequency of grade D by the total number of students: P(D) = \(\frac{6}{40} = \frac{3}{20}\)

Step 6: Calculate the probability of grade F

Divide the frequency of grade F by the total number of students: P(F) = \(\frac{2}{40} = \frac{1}{20}\) Now that we have calculated the probabilities for each grade, we can present the probability distribution as a table: $$ \begin{array}{c|c} \text{Grade} & \text{Probability} \\ \hline A & \frac{1}{10} \\ B & \frac{1}{4} \\ C & \frac{9}{20} \\ D & \frac{3}{20} \\ F & \frac{1}{20} \\ \end{array} $$

Key Concepts

Grade Distribution

In statistics, grade distribution refers to the way grades are spread out across different categories. This involves organizing grades into categories like A, B, C, D, and F. Each category includes the number of students that received that particular grade. Grade distribution helps in understanding how students in a class have performed overall. For instance, if the majority of the students scored a C, the grade distribution suggests a median performance. This can be significant for educators to identify trends, recognize areas where students excel, or need support.

Frequency Distribution

Frequency distribution is a key concept when analyzing data like grades. It refers to the representation of data showing how often each outcome occurs. In the context of grades, it deals with how many students received each type of grade.

• Frequency: The number of times a particular grade appears in the set of data. For example, 18 students received a C.
• Total Frequency: This is the sum of all individual frequencies, giving the total number of students or observations. In the given scenario, the total frequency is 40.

Understanding frequency distribution helps in visualizing how grades are spread across different categories. It also serves as a foundational step in calculating probabilities.

Probability Calculation

Probability calculation is the process used to determine the likelihood of each event occurring. In the context of grade distribution, it estimates how probable it is for a student to receive a particular grade.The probability of receiving a specific grade is calculated using the formula:\[ P( ext{grade}) = \frac{\text{Frequency of grade}}{\text{Total number of students}} \]This calculation involves dividing the frequency of a particular grade by the total number of students. For instance, the probability of getting an A is calculated as follows:- For grade A, with a frequency of 4: \[ P(A) = \frac{4}{40} = \frac{1}{10} \]Performing such calculations for all grades gives us a "probability distribution." This distribution helps us easily understand how likely each grade is, providing insights into the performance of the entire class.