Question

students at a large university have four places to get lunch: the cafeteria, the hut, the taco wagon, or the pizza place. an article in the school newsletter states that 70% of students prefer to get lunch in the cafeteria and the other three options are preferred equally. to investigate this claim, a random sample of 150 students is selected. 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:

Step1: Check Random Condition

A random sample of 150 students is selected. So, the random condition is met.

Step2: Check 10% Condition

Since the university has a large number of students (assume \(N\geq10\times150 = 1500\)), the 10% condition \(n\leq0.1N\) is met.

Step3: Check Large Counts Condition

• Proportion for cafeteria \(p_1 = 0.7\), expected count \(E_1=np_1=150\times0.7 = 105\)
• Proportion for each of the other three (hut, taco wagon, pizza place) \(p_2=\frac{1 - 0.7}{3}=0.1\)
• Expected count for each of the other three \(E_2=np_2=150\times0.1 = 15\)

Since \(E_1 = 105\geq5\) and \(E_2=15\geq5\), the large - counts condition is met.

Answer:

Yes, all of the conditions for inference are met.