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 failing the test are independent by filling out the blanks in the sentence below, rounding all probabilities to the nearest thousandth. since ( p )(did eat breakfast | fail) ( =square ) and ( p )(did eat breakfast) ( =square ), the two results are so the events are

Explanation:

Step1: Calculate \(P(\text{did eat breakfast}|\text{fail})\)

By the conditional - probability formula \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). Here, \(A\) is "did eat breakfast" and \(B\) is "fail". The number of students who failed is \(21 + 60=81\). The number of students who failed and did eat breakfast is \(21\). So \(P(\text{did eat breakfast}|\text{fail})=\frac{21}{81}\approx0.259\).

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

The total number of students is \(75 + 21+89 + 60=245\). The number of students who did eat breakfast is \(75 + 21 = 96\). So \(P(\text{did eat breakfast})=\frac{96}{245}\approx0.392\).

Since \(P(\text{did eat breakfast}|\text{fail})
eq P(\text{did eat breakfast})\), the two events are not independent.

Answer:

Since \(P(\text{did eat breakfast}|\text{fail}) = 0.259\) and \(P(\text{did eat breakfast})=0.392\), the two results are not equal so the events are not independent.