Answer:
The events are not independent
[tex]P(S) = 0.8[/tex]
[tex]P(Logo\ A) = 0.62[/tex]
Step-by-step explanation:
Given
--------------Logo A --- Logo B ---Total
Students ----- 67----------33--------100
Teachers ----- 11 ----- ----- 14 ----- 25
Total ----- ------78-----------47--------125
Required
Determine if being a student and preferring logo A are independent events
Let P(S) represent probability of being a student
Let P(Logo A) represent probability of preferring logo A
Let P(S n Logo A) represent probability of being a student and preferring logo A
To determine if they are independent, we have to calculate
P(S), P(Logo A) and P(S n Logo A)
P(S) = Total Students / Total Population
[tex]P(S) = 100/125 = 0.8[/tex]
P(Logo A) = Total that prefers Logo A / Total Population
[tex]P(Logo\ A) = 78/125 =0.624[/tex]
P(S n Logo A) = Number of students that prefer logo A / Total Population
[tex]P(S\ n\ Logo\ A) = 67/125 = 0.536[/tex]
If the events are independent, then the following condition must be satisfied
[tex]P(S\ n\ Logo\ A) = P(S) * P(Logo\ A)[/tex]
Substitute the values of P(S), P(Logo A) and P(S n Logo A)
[tex]0.536 = 0.8 * 0.536[/tex]
[tex]0.536 \neq 0.4288[/tex]
Since they are not equal, then the events are not independent
[tex]P(S) = 0.8[/tex]
[tex]P(Logo\ A) = 0.62[/tex]
