Question

the table summarizes the daily caffeine habits and majors of students at one university.
part a: determine p(no caffeine | stem major) and describe the event in everyday language. show all work. (5 points)
part b: are the events consuming no caffeine and a stem major approximately independent? use probabilities to justify the answer. (5 points)

Explanation:

Step1: Calculate the probability of STEM major

The probability of STEM major \(P(\text{STEM})\) is the sum of probabilities of STEM major with coffee, energy drink and no caffeine.
\(P(\text{STEM})=0.32 + 0.13+0.02=0.47\)

Step2: Calculate \(P(\text{no caffeine}|\text{STEM})\)

By the formula of conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\), here \(A\) is "no caffeine" and \(B\) is "STEM major". \(P(\text{no caffeine}\cap\text{STEM}) = 0.02\), \(P(\text{STEM})=0.47\)
\(P(\text{no caffeine}|\text{STEM})=\frac{0.02}{0.47}\approx0.0426\)
In everyday language, \(P(\text{no caffeine}|\text{STEM})\) represents the probability that a student has no - caffeine intake given that the student is a STEM major.

Step3: Calculate \(P(\text{no caffeine})\)

The probability of no caffeine \(P(\text{no caffeine})\) is the sum of probabilities of no - caffeine for STEM and non - STEM majors. \(P(\text{no caffeine})=0.02 + 0.09=0.11\)

Step4: Check for independence

Two events \(A\) (no caffeine) and \(B\) (STEM major) are independent if \(P(A|B)=P(A)\)
Since \(P(\text{no caffeine}|\text{STEM})\approx0.0426\) and \(P(\text{no caffeine}) = 0.11\), \(P(\text{no caffeine}|\text{STEM})
eq P(\text{no caffeine})\)

Answer:

Part A: \(P(\text{no caffeine}|\text{STEM})\approx0.0426\), which means the probability that a student has no - caffeine intake given that the student is a STEM major is approximately \(0.0426\)
Part B: The events “consuming no caffeine” and “a STEM major” are not independent. Because \(P(\text{no caffeine}|\text{STEM})\approx0.0426\) and \(P(\text{no caffeine})=0.11\), and \(P(\text{no caffeine}|\text{STEM})
eq P(\text{no caffeine})\)