Question

a two - way frequency table shows grades for students in college and students in high school:

	high school	college	total
gpa below 3.0	46	14	60
	60	40	100

based on this data, are \being in high school\ and \gpa above 3.0\ independent events?
yes, p(high school ∩ gpa above 3.0) = p(high school) · p(gpa above 3.0)
no, p(high school ∩ gpa above 3.0) = p(high school) · p(gpa above 3.0)
yes, p(high school ∩ gpa above 3.0) ≠ p(high school) · p(gpa above 3.0)
no, p(high school ∩ gpa above 3.0) ≠ p(high school) · p(gpa above 3.0)

Explanation:

Step1: Calculate P(high school)

$P(\text{high school})=\frac{60}{100} = 0.6$

Step2: Calculate P(GPA above 3.0)

$P(\text{GPA above 3.0})=\frac{40}{100}=0.4$

Step3: Calculate P(high school ∩ GPA above 3.0)

$P(\text{high school}\cap\text{GPA above 3.0})=\frac{14}{100} = 0.14$

Step4: Calculate P(high school)×P(GPA above 3.0)

$P(\text{high school})\times P(\text{GPA above 3.0})=0.6\times0.4 = 0.24$

Step5: Compare the two - probabilities

Since $0.14
eq0.24$, i.e., $P(\text{high school}\cap\text{GPA above 3.0})
eq P(\text{high school})\times P(\text{GPA above 3.0})$, the events are not independent.

Answer:

No, P(high school ∩ GPA above 3.0) ≠ P(high school)·P(GPA above 3.0)