2.2 Joint, Marginal and Conditional Relative Frequencies — Answers and Rubrics
These solutions explain the new practice tasks in this lesson. For simulation parts, a valid recorded run may differ from another learner’s run; the rubric checks the model, recorded work and interpretation.
Original generated practice for the Fall 2026 course. Return to the lesson and independent checks.
Calculate Relative Frequencies from a Café Table: worked answer
| Drink choice | Morning | Afternoon | Total |
|---|---|---|---|
| Tea | 18 | 42 | 60 |
| Coffee | 42 | 18 | 60 |
| Total | 60 | 60 | 120 |
Every one of the 120 orders is classified as tea or coffee and as morning or afternoon. The counts are exact for this day.
Part a
Construct a complete joint-relative-frequency table, including its marginal row/column totals. Identify tea-and-morning, tea overall, and afternoon overall, stating each denominator.
| Choice | Morning | Afternoon | Total |
|---|---|---|---|
| Tea | .15 | .35 | .50 |
| Coffee | .35 | .15 | .50 |
| Total | .50 | .50 | 1.00 |
Tea-and-morning is 18/120=.15; tea overall is 60/120=.50; afternoon overall is 60/120=.50. Every joint and marginal proportion uses grand total 120.
| Evidence for credit | Points |
|---|---|
| Constructs all four joint cells correctly. | 1 |
| Includes both sets of margins and total 1. | 1 |
| Names the requested proportions and denominator 120. | 1 |
Part b
Calculate the complete drink-choice distribution among afternoon orders and the complete time-of-day distribution among tea orders. Explain the restricted reference groups and why their totals are each 100%.
Among afternoon orders: tea 42/60=.70 and coffee 18/60=.30. Among tea orders: morning 18/60=.30 and afternoon 42/60=.70. The first denominator is the afternoon column total; the second is the tea row total. Each pair accounts for every order in its own restricted group and sums to 1.
| Evidence for credit | Points |
|---|---|
| Computes .70/.30 for afternoon drink choice. | 1 |
| Computes .30/.70 for time among tea orders. | 1 |
| Identifies column versus row reference groups and total 1. | 1 |
Compare and Justify a Café Claim: worked answer
| Drink choice | Morning | Afternoon | Total |
|---|---|---|---|
| Tea | 18 | 42 | 60 |
| Coffee | 42 | 18 | 60 |
| Total | 60 | 60 | 120 |
Every one of the 120 orders is classified as tea or coffee and as morning or afternoon. The counts are exact for this day.
Part a
Calculate and compare the complete drink-choice distributions in morning and afternoon. Determine whether time of day and drink choice are associated in these data.
Morning is 18/60=.30 tea and 42/60=.70 coffee; afternoon is .70 tea and .30 coffee. The conditional distributions differ, supporting association between time of day and drink choice for this café day.
| Evidence for credit | Points |
|---|---|
| Calculates both complete conditional distributions. | 1 |
| Compares them and makes the appropriate contextual association conclusion. | 1 |
Part b
Justify or reject the claim “Tea is a larger share of afternoon orders than morning orders.” Use calculated summary statistics, a numerical comparison and a precise contextual conclusion. Explain why equal overall tea/coffee totals do not settle this claim.
The claim is supported: tea is 70% of afternoon orders versus 30% of morning orders, a 40-percentage-point difference. Overall tea and coffee each total 60, but those marginal counts hide the different time-specific distributions. The conclusion describes this café day’s orders.
| Evidence for credit | Points |
|---|---|
| Uses .70 versus .30 and the correct time-specific denominators. | 1 |
| Supports the claim in the defined café population. | 1 |
| Explains why marginal totals are insufficient. | 1 |