Question

a 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.
answer attempt 1 out of 2
since p(did eat breakfast)×p(pass) = □ and p(did eat breakfast and pass) = □, the two results are so the events are.

Explanation:

Step1: Calculate total number of students

Total students \(=66 + 33+76 + 38=213\)

Step2: Calculate \(P(\text{did eat breakfast})\)

Number of students who ate breakfast \(=66 + 33 = 99\). So \(P(\text{did eat breakfast})=\frac{99}{213}\approx0.465\)

Step3: Calculate \(P(\text{pass})\)

Number of students who passed \(=66+76 = 142\). So \(P(\text{pass})=\frac{142}{213}\approx0.667\)

Step4: Calculate \(P(\text{did eat breakfast})\times P(\text{pass})\)

\(P(\text{did eat breakfast})\times P(\text{pass})=0.465\times0.667 = 0.310\)

Step5: Calculate \(P(\text{did eat breakfast and pass})\)

Number of students who ate breakfast and passed \(=66\). So \(P(\text{did eat breakfast and pass})=\frac{66}{213}\approx0.310\)

Answer:

Since \(P(\text{did eat breakfast})\times P(\text{pass}) = 0.310\) and \(P(\text{did eat breakfast and pass})=0.310\), the two results are equal so the events are independent.