a Which column of the given contingency table represents the number of students who took the preparatory class?
B
b How many students neither took the preparatory classes nor passed the exams?
C
c Calculate the probability that a student passed the exam given that he or she did not take the preparatory class.
A
a ≈ 0.82
B
b ≈ 0.35
C
c Yes, see solution.
Practice makes perfect
a We are given information about the number of nursing students who took preparatory class before taking board exams and the number of students who passed on the first attempt. Let's take a look at the given contingency table.
Preparatory Class
No Preparatory Class
Totals
Passed Exams
14
11
25
Did Not Pass Exams
3
6
9
Totals
17
17
34
We want to find the probability that a student passed the exams given he or she took a preparatory class. Let m denote the number of students that took the class and passed the exams and n denote the number of all students that took the class.
P(passed | class) = m/n
The first column of the data represents the total number of students who took the preparatory class, 17. We can also see that the number of those students who passed the exams on their first attempt is 14. Finally, let's substitute these values to calculate the probability.
b Just as in Part A, let's take a look at the given data.
Preparatory Class
No Preparatory Class
Totals
Passed Exams
14
11
25
Did Not Pass Exams
3
6
9
Totals
17
17
34
To find the probability, first let n denote the number of all students that did not take the preparatory class. We will also let m be the number of those students who neither took the class nor passed the exams.
P(not passed | no class) = m/n
Note that the second column of the contingency table shows the total number of students who did not take the preparatory classes, 17. The second row of this column represents the number of students from this group who did not pass the exams.
P(not passed | no class) = 6/17 ≈ 0.35
c To answer the question of whether taking the preparatory class is a good decision, we will first calculate the probability that a nursing student passed the exams and did not attend the class. Then we will compare this result with the result from Part A.
Preparatory Class
No Preparatory Class
Totals
Passed Exams
14
11
25
Did not Pass Exams
3
6
9
Totals
17
17
34
Similar to Parts A and B, let's use the given table to calculate the probability. Note that there are 11 students that passed the exams after not attending the class. Additionally, there are 17 students in total who did not take the class. Let's calculate the probability!
P(passed | no class) = 11/17 ≈ 0.647
Now we can see that about 6 out of 10 students who do not take the class pass the exam. From Part A, we can conclude that about 8 out of 10 students pass the exams after taking the class. Therefore, the class appears to be beneficial and the student makes a good decision.
