Suppose a survey records whether $\mathrm{100}$ students prefer coffee or tea ($\mathrm{Coffee:}60$, $\mathrm{Tea:}40$) and whether they are undergraduate or graduate students ($\mathrm{Undergraduate:}70$, $\mathrm{Graduate:}30$). Which two-way table correctly displays this data?

• A. $\mathrm{Undergraduate:}\mathrm{Coffee}35,\mathrm{Tea}35$; $\mathrm{Graduate:}\mathrm{Coffee}25,\mathrm{Tea}5$
• B. $\mathrm{Undergraduate:}\mathrm{Coffee}42,\mathrm{Tea}28$; $\mathrm{Graduate:}\mathrm{Coffee}18,\mathrm{Tea}12$
• C. $\mathrm{Undergraduate:}\mathrm{Coffee}30,\mathrm{Tea}40$; $\mathrm{Graduate:}\mathrm{Coffee}30,\mathrm{Tea}0$
• D. $\mathrm{Undergraduate:}\mathrm{Coffee}60,\mathrm{Tea}10$; $\mathrm{Graduate:}\mathrm{Coffee}0,\mathrm{Tea}30$

Answer: (B)

Verified step by step guidance

1. Step 1: Understand the problem context. We have two categorical variables: beverage preference (Coffee or Tea) and student status (Undergraduate or Graduate). The total number of students is 100, with 60 preferring coffee and 40 preferring tea. Also, 70 are undergraduates and 30 are graduates.
2. Step 2: Recognize that the two-way table should display counts that sum correctly across both rows and columns. Specifically, the sum of coffee drinkers across undergraduates and graduates should be 60, and the sum of tea drinkers should be 40. Similarly, the sum of undergraduates across coffee and tea should be 70, and the sum of graduates should be 30.
3. Step 3: Set up the two-way table with variables: let \(U_C\) be the number of undergraduates who prefer coffee, \(U_T\) undergraduates who prefer tea, \(G_C\) graduates who prefer coffee, and \(G_T\) graduates who prefer tea. The table looks like this: \[ \begin{array}{c|cc|c} & \text{Coffee} & \text{Tea} & \text{Total} \\ \hline \text{Undergraduate} & U_C & U_T & 70 \\ \text{Graduate} & G_C & G_T & 30 \\ \hline \text{Total} & 60 & 40 & 100 \\ \end{array} \]
4. Step 4: Use the marginal totals to form equations: \[ U_C + G_C = 60 \quad \text{(total coffee drinkers)} \] \[ U_T + G_T = 40 \quad \text{(total tea drinkers)} \] \[ U_C + U_T = 70 \quad \text{(total undergraduates)} \] \[ G_C + G_T = 30 \quad \text{(total graduates)} \]
5. Step 5: Check each proposed two-way table against these equations to verify if the sums match the marginal totals. The correct table will satisfy all four equations simultaneously.