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

Let’s solve the problem step by step using the two-way table provided.

First, let’s write down the full table clearly:

| | Passed | Failed | Total |
|-----------|--------|--------|-------|
| Studied | 12 | 3 | 15 |
| Did Not Study | ? | 14 | 15 |
| Total | ? | ? | ? |

Wait — we need to fill in the missing values first.

We know:
- “Studied” row total = 15 → 12 passed + 3 failed = 15 ✔️
- “Did Not Study” row total = 15 → so if 14 failed, then passed = 15 - 14 = 1
- Now, total who passed = 12 (studied) + 1 (did not study) = 13
- Total who failed = 3 (studied) + 14 (did not study) = 17
- Grand total = 15 (studied) + 15 (did not study) = 30

So completed table:

| | Passed | Failed | Total |
|-----------|--------|--------|-------|
| Studied | 12 | 3 | 15 |
| Did Not Study | 1 | 14 | 15 |
| Total | 13 | 17 | 30 |

Now, answer each question one by one.

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Question 1: How many students studied?

Look at the “Studied” row total → 15

✔ Answer: 15

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Question 2: How many students failed?

Look at the “Failed” column total → 17

✔ Answer: 17

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Question 3: How many students did not study?

“Did Not Study” row total → 15

✔ Answer: 15

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Question 4: What percentage of students did not study?

Total students = 30
Students who did not study = 15

Percentage = (15 ÷ 30) × 100 = 0.5 × 100 = 50%

✔ Answer: 50%

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Question 5: What percentage of students passed the test?

Total passed = 13
Total students = 30

Percentage = (13 ÷ 30) × 100 ≈ 43.33%

But since this is likely for middle/high school, we can leave it as a fraction or round appropriately. Let’s keep it exact unless told otherwise.

Actually, let’s check: 13/30 = 0.4333... → 43.3% (rounded to nearest tenth)

But maybe they want exact fraction? The question says “percentage”, so decimal is fine.

✔ Answer: Approximately 43.3%

Wait — let me double-check: 13 out of 30.

Yes, 13 ÷ 30 = 0.4333… → 43.3%

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Question 6: Out of the students that passed the test, what percentage had studied?

This is a conditional probability: given that a student passed, what % studied?

Number who passed AND studied = 12
Total who passed = 13

So percentage = (12 ÷ 13) × 100 ≈ 92.3%

✔ Answer: Approximately 92.3%

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Question 7: What is the relative frequency of students who studied and passed the test?

Relative frequency = part / whole

Here, “whole” is total number of students = 30
Part = students who studied AND passed = 12

So relative frequency = 12 / 30 = 0.4

✔ Answer: 0.4

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Question 8: What is the relative frequency of students who failed the test?

Total who failed = 17
Total students = 30

Relative frequency = 17 / 30 ≈ 0.5667

But again, maybe leave as fraction or decimal? Since Q7 was 0.4, probably decimal is fine.

17 ÷ 30 = 0.5666... → rounds to 0.567 if to three decimals, but often in stats, we keep as fraction or simplify.

Wait — relative frequency is usually expressed as a decimal between 0 and 1.

So 17/30 is exact, but if decimal is needed: ≈ 0.567

But let’s see what’s standard. In many textbooks, they accept fractions or decimals.

Since Q7 was 12/30 = 0.4, which is exact, here 17/30 is not terminating, so perhaps write as fraction? But the question doesn’t specify.

Looking back at context — it’s a worksheet, likely expects decimal.

But to be precise, let’s calculate:

17 ÷ 30 = 0.5666... → so 0.567 if rounded to three decimal places.

However, sometimes they want exact value. Let me check common practice.

In two-way tables, relative frequency is often reported as decimal rounded to 3 places or as fraction.

But since Q7 is 12/30 = 2/5 = 0.4, maybe they expect simplified fraction or decimal.

I think for consistency, since 12/30 reduces to 0.4, and 17/30 doesn't reduce nicely, we’ll go with decimal approximation.

But wait — perhaps I should present both? No, the instruction is to give final answer.

Let me re-read: “What is the relative frequency...”

In statistics, relative frequency is count divided by total, so 17/30 is exact, but often written as decimal.

Given that, and since Q7 is 0.4, I'll do same for Q8: 17/30 ≈ 0.567

But let's confirm calculation:

17 ÷ 30 = ?

30 * 0.5 = 15 → remainder 2 → 2/30 = 0.0666... → so 0.5666... → yes, 0.567 when rounded to three decimal places.

✔ Answer: 0.567 (or approximately 0.567)

But to be safe, maybe write as fraction? The problem doesn't specify.

Looking at the image, it says "relative frequency", and in such contexts, decimal is common.

I think 0.567 is acceptable.

Alternatively, leave as 17/30? But Q7 was 12/30=0.4, so likely decimal expected.

I'll go with 0.567.

Wait — actually, in many curricula, they might expect the exact fraction. But since 12/30 was simplified to 0.4, which is exact, and 17/30 is not, perhaps write as decimal.

Another thought: relative frequency can be left as fraction. But let's see the level — middle/high school, so decimal is fine.

I think I'm overthinking. Let's proceed.

Final decision: for Q7, 12/30 = 0.4; for Q8, 17/30 ≈ 0.567

But to match precision, perhaps report both as decimals with three places? 0.400 and 0.567? But 0.4 is fine.

Actually, 12/30 is exactly 0.4, so no issue.

For Q8, 17/30 is approximately 0.567.

✔ So Answer: 0.567

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Now, summarizing all answers:

1. 15
2. 17
3. 15
4. 50%
5. 43.3% (or 43.33%, but 43.3 is sufficient)
6. 92.3%
7. 0.4
8. 0.567

Wait — for Q5 and Q6, percentages, should I round to nearest whole number? The problem doesn't specify.

In Q4, 50% is exact.

Q5: 13/30 = 43.333...% → typically rounded to one decimal place → 43.3%

Q6: 12/13 ≈ 92.307...% → 92.3%

That seems consistent.

Now, for the final answer, since the user didn't specify which question, but the image shows questions 1 through 8, and the assistant is to solve the problem, I think we need to provide all answers.

But the instruction says: "Solve the problem accurately." and "provide the final correct answer".

Looking back, the user uploaded an image with multiple questions, so likely expects all answers.

But the response format says: "Final Answer: [Provide only the final correct answer]"

Hmm, this is ambiguous. Perhaps list all answers.

Maybe the "problem" refers to the entire set.

To be safe, I'll list all answers in order.

But let me check the initial request: "Solve the problem accurately." and the image has 8 questions.

Perhaps the assistant should output all answers.

Given that, I'll compile them.

Final Answers:

1. 15
2. 17
3. 15
4. 50%
5. 43.3%
6. 92.3%
7. 0.4
8. 0.567

But for Q8, is 0.567 precise enough? Or should I use more decimals?

17/30 = 0.5666..., so to three decimal places, it's 0.567.

Some might prefer fraction: 17/30.

But since Q7 is 0.4, which is decimal, I'll stick with decimal.

Alternatively, write as fractions for accuracy.

Let me reconsider.

In educational contexts, especially for relative frequency, they often accept fractions.

For example, Q7: 12/30 = 2/5, but they wrote 0.4 in my earlier thought.

Actually, in the table, numbers are integers, so fractions are exact.

Perhaps better to use fractions where possible.

Q7: 12/30 = 2/5 or 0.4 — both fine, but 0.4 is simpler.

Q8: 17/30 cannot be simplified, so either 17/30 or 0.567.

I think for consistency, since the total is 30, and counts are small, fractions might be preferred.

But let's see the worksheet style — it says "relative frequency", and in many cases, it's decimal.

I recall that in AP Stats, they use decimals.

For middle/high school, decimals are common.

I'll go with decimals for Q7 and Q8.

So:

Q7: 0.4
Q8: 0.567 (rounded to three decimal places)

To be precise, let's calculate 17÷30:

30 * 0.566 = 16.98, close to 17, difference 0.02, so 0.566 + 0.02/30 ≈ 0.566 + 0.00067 = 0.56667, so 0.567 when rounded.

Yes.

Now, for the final answer section, since there are multiple questions, I'll list them numerically.

Final Answer:
1. 15
2. 17
3. 15
4. 50%
5. 43.3%
6. 92.3%
7. 0.4
8. 0.567

But to make it clean, perhaps write as a list.

Since the user might expect boxed answers, but the format says "Final Answer: [provide only the final correct answer]", implying one answer, but that doesn't make sense for multiple questions.

Perhaps the "problem" is to interpret the table, but I think providing all answers is necessary.

Another idea: maybe the assistant is supposed to solve based on the visible part, but the image shows up to Q8.

I think it's safe to provide all.

Perhaps in the context, the main task is to understand the table, but the questions are given.

I'll proceed with listing all answers.

So, after careful step-by-step calculation, here are the answers:

Final Answer:
1. 15
2. 17
3. 15
4. 50%
5. 43.3%
6. 92.3%
7. 0.4
8. 0.567