Question

is being under the age of 12 independent of visiting the inventions exhibit at this museum? justify the conclusion.
no, because 0.074≠(0.368)(0.321)
yes, because 0.074=(0.368)(0.321)
no, because 0.118≠(0.368)(0.321)
yes, because 0.118=(0.368)(0.321)

	under 12	12 - 25	over 25
modern art	5	8	17
inventions	14	28	19
aerospace	16	7	9

Explanation:

Step1: Calculate total number of visitors

First, sum all the values in the table. $12 + 20+35 + 17+8 + 5+19+28+14+9+7+16=180$.

Step2: Calculate probability of being under 12

The number of visitors under 12 is $35 + 5+14+16 = 70$. The probability $P(\text{under }12)=\frac{70}{180}\approx0.389$.

Step3: Calculate probability of visiting inventions exhibit

The number of visitors to inventions exhibit is $19 + 28+14 = 61$. The probability $P(\text{inventions})=\frac{61}{180}\approx0.339$.

Step4: Calculate probability of being under 12 and visiting inventions exhibit

The number of visitors who are under 12 and visit inventions exhibit is 14. The probability $P(\text{under }12\text{ and inventions})=\frac{14}{180}\approx0.078$.

Step5: Check for independence

If two events $A$ (being under 12) and $B$ (visiting inventions exhibit) are independent, then $P(A\cap B)=P(A)\times P(B)$. $P(A)\times P(B)=\frac{70}{180}\times\frac{61}{180}=\frac{4270}{32400}\approx0.132
eq0.078 = P(A\cap B)$.

Answer:

No, because $0.078
eq(0.389)(0.339)$ (approximate values used for the non - equality check based on the calculated probabilities).