group of students at a high school took a standardized test. The number of students who passed or failed the exam is broken down by those who did or did not eat breakfast in the following table. Determine whether eating breakfast and passing the test are independent by filling out the blanks in the sentence below, rounding all probabilities to the nearest thousandth Passed Failed Did Eat Breakfast 237 Didn't Eat Breakfast 46 14
To determine whether eating breakfast and passing the test are independent, we need to calculate the probability of passing the test given that the student ate breakfast, and compare it to the probability of passing the test overall.
The total number of students who passed the test is 237 + 46 = 283. The total number of students who did not eat breakfast is 46 + 14 = 60. The total number of students who ate breakfast is 237 + 14 = 251.
The probability of passing the test overall is 283/300 = 0.943.
The probability of passing the test given that the student ate breakfast is 237/251 = 0.944.
Since the probability of passing the test given that the student ate breakfast is very close to the overall probability of passing the test, we can conclude that eating breakfast and passing the test are independent.