Question

a museum curator tracked the ages of visitors who frequented different exhibits and displayed the data in a table. is being under the age of 12 independent of visiting the inventions exhibit at this museum? justify the conclusion. under 12 12 - 25 over 25 dinosaurs 35 20 12 modern art 5 8 17 inventions 14 28 19 aerospace 16 7 9 yes, because 0.118=(0.368)(0.321) no, because 0.118≠(0.368)(0.321) yes, because 0.074=(0.368)(0.321) no, because 0.074≠(0.368)(0.321)

Explanation:

Step1: Calculate total number of visitors

First, sum up 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 \(35 + 5+14+16=70\). So the probability \(P(\text{under }12)=\frac{70}{180}\approx0.389\)

Step3: Calculate probability of visiting inventions exhibit

The number of visitors to the inventions exhibit is \(19 + 28+14 = 61\). So 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 visited the inventions exhibit is 14. So the probability \(P(\text{under }12\text{ and inventions})=\frac{14}{180}\approx0.078\)

Step5: Check for independence

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

Answer:

No, because \(0.078
eq(0.389)(0.339)\) (rounded - off values used for simplicity of showing non - equality, similar to the values in the problem setup)