One Categorical Variable: Tables and Summaries — Worked Solutions
Worked Solutions
Compare your reasoning with the worked solutions after attempting the questions. A different valid representation or explanation can earn credit when it satisfies the stated criteria.
Original generated practice for the Fall 2026 course. Return to the lesson and independent checks.
Read the frequency: worked answer
A sample of 20 customers chooses one drink: tea 8, coffee 7, water 5.
Part a
What does the tea frequency count? State its relative frequency and the denominator.
It counts 8 customers in this sample who chose tea. Relative frequency is 8/20=0.40, using all 20 sampled customers as the denominator.
| Evidence for credit | Points |
|---|---|
| Contextual count and correct proportion/denominator. | 1 |
Three ways to say the same share: worked answer
Five of 25 sampled customers choose water.
Part a
Express this share as a relative frequency, percentage, and count-to-total ratio. Explain the common meaning.
5/25=0.20=20%; the ratio is 5:25 or 1:5. Each means one-fifth of these sampled customers chose water.
| Evidence for credit | Points |
|---|---|
| All three equivalent summaries and common contextual meaning. | 1 |
A claim needs the right total: worked answer
| Meal | Count |
|---|---|
| Wrap | 9 |
| Soup | 6 |
| Salad | 5 |
Part a
Describe the most common choice with a numerical summary. Evaluate the claim “most diners chose wraps.”
Wrap is the most common single category, with 9 of 20 sampled diners or 45%. It is not a majority: 45% is below half, so “most” meaning more than half is unsupported.
| Evidence for credit | Points |
|---|---|
| Largest category and 45% contextual summary. | 1 |
| Distinguishes modal category from majority. | 1 |
Compare shares: worked answer
Sample A has 12 cycle commuters out of 30 people; sample B has 20 out of 80.
Part a
Describe the cycle shares in both samples and explain why the larger count does not show the larger share.
A has 12/30=40% cycling, while B has 20/80=25%. B has more observed cyclists but a smaller proportion because its total is larger.
| Evidence for credit | Points |
|---|---|
| Correct two proportions and explanation of differing totals. | 1 |
Organize a new mode survey: worked answer
Twenty park visitors report their usual trip mode, once each: Bus, Bike, Car, Bus, Walk, Bus, Bike, Car, Bus, Bike, Walk, Bus, Car, Bike, Bus, Walk, Bus, Bike, Car, Bus.
Part a
Construct a frequency table with all four categories and a total row. Explain what each frequency counts.
Bus 8, Bike 5, Car 4, Walk 3; total 20. Each frequency counts sampled park visitors reporting that category. The categories account for all 20 observations once.
| Evidence for credit | Points |
|---|---|
| All four counts and total. | 1 |
| Contextual unit/category explanation. | 1 |
Part b
Construct the corresponding relative frequency table. Check its total and explain the denominator.
Bus 0.40, Bike 0.25, Car 0.20, Walk 0.15; total 1. Each count is divided by all 20 sampled visitors.
| Evidence for credit | Points |
|---|---|
| All proportions and total 1. | 1 |
| Correct common denominator. | 1 |
Part c
Describe the bike share as a fraction/proportion, percentage, and count-to-total ratio.
5/20=0.25=25%, and 5:20=1:4. These all describe one-quarter of the sampled visitors reporting bike.
| Evidence for credit | Points |
|---|---|
| All equivalent representations and context. | 1 |
Part d
Describe two useful features of the distribution with numerical summaries. Assess “a majority arrived by bus” and “equal numbers biked and walked.”
Bus is the largest category at 8/20=40%, and bike plus walk is also 8/20=40%. Bus is not a majority because 40% is below 50%. Bike and walk are not equally frequent: 5 versus 3 visitors, or 25% versus 15%.
| Evidence for credit | Points |
|---|---|
| Two numerically grounded contextual features. | 1 |
| Both claims assessed from correct counts/proportions. | 1 |