Question

a school administrator believes that 80% of the students at his large school arrive by bus, 10% drive themselves, and 10% are dropped off by an adult. to investigate this belief, the administrator stands at the school entrance and asks the first 60 students who come through the door how they came to school. he would like to know if the data provide convincing evidence that the distribution of transportation method differs from what he believes it to be. are the conditions for inference met?
no, the random condition is not met.
no, the 10% condition is not met.
no, the large counts condition is not met.
yes, all of the conditions for inference are met.

Explanation:

To perform a chi - square goodness - of - fit test (which is likely the inference procedure here), the following conditions are required:

• Random Condition: The sample should be a random sample from the population. In this case, the administrator is using a convenience sample (the first 60 students who come through the door). This is not a random sample of all students at the school. A random sample would involve a method like randomly selecting students from a list of all students (e.g., using a random number generator to pick student IDs).
• 10% Condition: Since the school is large, if we assume the population of students \(N\) is such that \(n = 60\leq0.1N\) (where \(n\) is the sample size), this condition is likely met. But the non - random nature of the sample makes this less relevant in the context of proper inference.
• Large Counts Condition: The expected counts \(E=np\). For the bus category: \(E_1 = 60\times0.8=48\), for the drive - themselves category: \(E_2=60\times0.1 = 6\), for the dropped - off category: \(E_3=60\times0.1=6\). Since all expected counts \(E_i\geq5\), the large counts condition is met. But again, the non - random sample overshadows this.

Since the random condition (a fundamental condition for valid inference) is not met, we cannot proceed with a valid inference.

Answer:

No, the random condition is not met.