Two-Way Frequency Tables Notes and Worksheets - Lindsay Bowden - Free Printable

Let’s solve this step by step.

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Given:

- 20 freshmen (9th graders) and 20 sophomores (10th graders) surveyed → Total students = 40.
- 13 sophomores said they had dogs.
- 9 freshmen had cats.

We are to fill out the two-way frequency table:

| | Freshmen | Sophomores | Total |
|-----------|----------|------------|-------|
| Dogs | ? | 13 | ? |
| Cats | 9 | ? | ? |
| Total | 20 | 20 | 40 |

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Step 1: Fill in the missing values

#### For Freshmen:
- Total freshmen = 20
- Freshmen with cats = 9
→ So, freshmen with dogs = 20 - 9 = 11

#### For Sophomores:
- Total sophomores = 20
- Sophomores with dogs = 13
→ So, sophomores with cats = 20 - 13 = 7

#### Now fill row totals:
- Dogs total = 11 (freshmen) + 13 (sophomores) = 24
- Cats total = 9 (freshmen) + 7 (sophomores) = 16

#### Column totals already given: Freshmen = 20, Sophomores = 20, Total = 40

✔ Completed Table:

| | Freshmen | Sophomores | Total |
|-----------|----------|------------|-------|
| Dogs | 11 | 13 | 24 |
| Cats | 9 | 7 | 16 |
| Total | 20 | 20 | 40 |

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Now answer the questions:

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1. How many 10th graders have cats?
→ 10th graders = Sophomores → Cats = 7

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2. How many students own dogs?
→ Total in “Dogs” row = 24

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3. How many freshmen own dogs?
→ Freshmen with dogs = 11

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4. What percentage of students own cats?
→ Total cat owners = 16
→ Total students = 40
→ Percentage = (16 / 40) × 100 = 40%

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5. What percentage of the students surveyed were in 9th grade?
→ 9th graders = Freshmen = 20
→ Total students = 40
→ Percentage = (20 / 40) × 100 = 50%

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6. What is the relative frequency of 9th grade dog owners?
→ Relative frequency = (Number of 9th grade dog owners) / (Total students)
= 11 / 40 = 0.275 or 27.5%

*(Note: Sometimes “relative frequency” can mean within a group — but since it says “of students”, we use total students as denominator.)*

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7. Out of the sophomores, what is the relative frequency of owning a cat?
→ Sophomores who own cats = 7
→ Total sophomores = 20
→ Relative frequency = 7 / 20 = 0.35 or 35%

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8. Is there an association between grade levels and pet choices?

Let’s compare proportions:

- Freshmen (9th):
- Dogs: 11/20 = 55%
- Cats: 9/20 = 45%

- Sophomores (10th):
- Dogs: 13/20 = 65%
- Cats: 7/20 = 35%

→ Sophomores are more likely to own dogs than freshmen (65% vs 55%), and less likely to own cats (35% vs 45%).

This suggests there may be a weak association: 10th graders slightly prefer dogs, 9th graders slightly prefer cats.

✔ Answer: Yes, there appears to be a slight association — sophomores are more likely to own dogs, and freshmen more likely to own cats.

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9. Which pet choice is more popular?
→ Dogs: 24 students
→ Cats: 16 students
→ Dogs are more popular

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10. Is this statement always true: A 10th grader is more likely to own a dog?

→ From our data: 65% of sophomores own dogs vs 55% of freshmen → so *in this sample*, yes.

BUT — the question asks “is this statement always true?”

→ No. This is based on one small survey of 40 students. It might not hold for all schools, regions, or populations. Also, “always” implies universally true — which we cannot conclude from a single sample.

✔ Answer: No, this statement is not always true — it’s only true for this specific sample. Generalizing requires more data.

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✔ Final Answers Summary:

1. 7
2. 24
3. 11
4. 40%
5. 50%
6. 0.275 or 27.5%
7. 0.35 or 35%
8. Yes, there is a slight association — sophomores are more likely to own dogs.
9. Dogs
10. No, it is not always true — this is based on a sample and may not generalize.

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Let me know if you’d like to visualize this or calculate conditional probabilities!