A school's administrators are concerned about absenteeism at their school
and the e�ect it has on students' grades. The following data were compiled, based on a sample of students.
Student's Absent Medium Absent
Grade Infrequently Absent Frequently
Low 1 9 8
Medium 6 35 7
High 8 9 2
a. What is the probability that a randomly selected student will have a low grade?
This would be 1/15 + 9/53 + 8/17 right?
b. What is the probability that a randomly selected student absent frequently?
8/18 + 7/48 + 2/19
c. Are grades and frequency in absent independent events? Why?
Not sure but i think they are dependent
d. Knowing that a randomly selected student has high grade, what is the probability that student is absent frequently?
2/19
e. What is the probability that a randomly selected student has either high grade or absent frequently?
I don't know how to calculate this
a. total of low grade students/grand total of students = 18/85
b. total of frequently absent students/grand total of students
c. Yes,try a Chi-squared (X^2) test to show why
d. Yes!
e. Either-or probabilities are found by adding the probabilities of the individual events. Total number of high grade/grand total + total number of absent frequently/grand total
a. To find the probability that a randomly selected student will have a low grade, we need to sum the probabilities of the student being in the "Low" grade category across all levels of absenteeism.
Looking at the table, we see that there are a total of 1 + 6 + 8 = 15 students in the "Low" grade category. Therefore, the probability of randomly selecting a student with a low grade is 15/total number of students.
b. Similarly, to find the probability that a randomly selected student absent frequently, we need to sum the probabilities of the student being in the "Frequently" absent category across all levels of grade.
From the table, we see that there are a total of 8 + 7 + 2 = 17 students in the "Frequently" absent category. Therefore, the probability of randomly selecting a student who is absent frequently is 17/total number of students.
c. To determine if grades and frequency of absenteeism are independent, we need to compare the joint probability of a student having a certain grade and being absent at a certain frequency to the product of the individual probabilities of having the grade and being absent at that frequency.
In this case, we can calculate the joint probability for each grade and frequency combination, and then compare it to the product of the individual probabilities.
For example, for the "Low" grade and "Infrequently" absent category, the joint probability is 1/total number of students, the individual probability of having a "Low" grade is 15/total number of students, and the individual probability of being "Infrequently" absent is 1/total number of students.
If the joint probability equals the product of the individual probabilities for all grade and frequency combinations, then grades and frequency of absenteeism can be considered as independent events. Otherwise, they are dependent events.
d. To find the probability that a randomly selected student with a high grade is absent frequently, we can calculate the joint probability of having a high grade and being absent frequently, and divide it by the probability of having a high grade.
From the table, we see that the joint probability of having a high grade and being absent frequently is 2/total number of students, and the probability of having a high grade is (8 + 9 + 2)/total number of students.
Therefore, the probability that a randomly selected student with a high grade is absent frequently is (2/total number of students) divided by ((8 + 9 + 2)/total number of students).
e. To find the probability that a randomly selected student has either a high grade or is absent frequently, we need to calculate the probabilities of each event individually and then add them together.
The probability of a randomly selected student having a high grade is (8 + 9 + 2)/total number of students, and the probability of a randomly selected student being absent frequently is 17/total number of students.
Therefore, the probability of a randomly selected student having either a high grade or being absent frequently is (8 + 9 + 2)/total number of students + 17/total number of students.