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 23 7 Didn't Eat Breakfast 46 14
To determine if eating breakfast and passing the test are independent, we need to compare the probability of passing the test for students who ate breakfast to the overall probability of passing the test, and compare that to the probability of passing the test for students who did not eat breakfast to the overall probability of passing the test.
The overall probability of passing the test is the total number of students who passed divided by the total number of students: (23 + 7 + 46 + 14) / (23 + 7 + 46 + 14) = 0.622.
The probability of passing the test for students who ate breakfast is the number of students who passed and ate breakfast divided by the total number of students who ate breakfast: 23 / (23 + 7) = 0.767.
The probability of passing the test for students who did not eat breakfast is the number of students who passed and did not eat breakfast divided by the total number of students who did not eat breakfast: 46 / (46 + 14) = 0.767.
Since the probability of passing the test for students who ate breakfast is not equal to the overall probability of passing the test (0.767 ≠ 0.622) and the probability of passing the test for students who did not eat breakfast is not equal to the overall probability of passing the test (0.767 ≠ 0.622), we can conclude that eating breakfast and passing the test are dependent.