Question

The relative frequency table describes whether a group of students eat breakfast on school days and non-school days.


Eat Breakfast Do Not Eat Breakfast
School Day 0.18 0.49
Not a School Day 0.09 0.24
For these students, are eating breakfast and school day approximately independent? Justify the answer with probabilities.
The events are approximately independent because 0.269 ≠ 0.670.
The events are not independent because 0.269 ≠ 0.670.
The events are approximately independent because 0.269 ≈ 0.270.
The events are not independent because 0.269 ≈ 0.270.

Answer 1

To determine if the events of eating breakfast and school day are approximately independent for the given group of students, we can check if the joint probability of these events is approximately equal to the product of their marginal probabilities.

The joint probability of eating breakfast on a school day is 0.18, and the marginal probability of eating breakfast overall is the sum of the probabilities of eating breakfast on a school day and not on a school day, which is 0.18 + 0.09 = 0.27.

The marginal probability of a school day is the sum of the probabilities of eating breakfast on a school day and not eating breakfast on a school day, which is 0.18 + 0.49 = 0.67.

Therefore, the product of these marginal probabilities is approximately 0.27 x 0.67 = 0.181, which is quite close to the joint probability of eating breakfast on a school day (0.18). Hence, we can say that the events are approximately independent because the difference between the joint probability and the product of marginal probabilities is not significant.

Therefore, the correct answer is (c) The events are approximately independent because 0.269 ≈ 0.270.