Two-way frequency tables homework key - Math 19b - Studocu - Free Printable

Let’s solve each part step by step.

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Problem 1: Swimming Lessons

We are given a table with data about students in grades 7, 8, and 9 — whether they can swim or cannot swim.

First, let’s complete the left table (frequency table):

| Grade | Can Swim | Cannot Swim | Total |
|---------|----------|-------------|-------|
| Grade 7 | 120 | 60 | ? |
| Grade 8 | 168 | 11 | ? |
| Grade 9 | 172 | 3 | ? |
| Total | ? | ? | ? |

→ Grade 7 total = 120 + 60 = 180
→ Grade 8 total = 168 + 11 = 179
→ Grade 9 total = 172 + 3 = 175
→ Total “Can Swim” = 120 + 168 + 172 = 460
→ Total “Cannot Swim” = 60 + 11 + 3 = 74
→ Grand total = 180 + 179 + 175 = 534 (or 460 + 74 = 534) ✔

Now, the right table (relative frequency table) — we convert each number to a percentage of the grand total (534).

But note: The handwritten answers show percentages that add up to 100% across rows and columns — so it seems they’re using row relative frequencies for the grade rows, and column totals as percentages of overall total? Actually, looking at the right table:

It shows:
- Grade 7: 22%, 11%, 33% → 22+11=33 → this is row % (of grade 7 total)
Wait — 120/180 ≈ 66.7%, not 22%. So maybe it’s column relative frequency? Let’s check:

Total “Can Swim” = 460 → 120/460 ≈ 26.1% — not 22%.

Actually, looking at the handwritten answer:
Grade 7 Can Swim = 22% → 120 / 534 ≈ 0.2247 → 22% ✔
Grade 7 Cannot Swim = 60 / 534 ≈ 0.1124 → 11% ✔
Grade 7 Total = 180 / 534 ≈ 0.337 → 33% ✔

So yes — all values in the right table are percentages of the grand total (534).

Let’s verify one more:
Grade 8 Can Swim = 168 / 534 ≈ 0.3146 → 31% ✔
Grade 8 Cannot Swim = 11 / 534 ≈ 0.0206 → 2% ✔
Grade 8 Total = 179 / 534 ≈ 0.335 → 33% ✔

Grade 9 Can Swim = 172 / 534 ≈ 0.322 → 32% ✔
Grade 9 Cannot Swim = 3 / 534 ≈ 0.0056 → 1% ✔
Grade 9 Total = 175 / 534 ≈ 0.3277 → 33% ✔

Column Totals:
Can Swim total = 460 / 534 ≈ 0.861 → but written as 85%? Wait — 460 ÷ 534 = 0.8614 → should be 86%, but they wrote 85%. Hmm.

Wait — actually, if you add the column percentages:
22% + 31% + 32% = 85% → that’s what they have. But 22+31+32=85, which matches.

Similarly, Cannot Swim: 11% + 2% + 1% = 14% ✔
Total: 33% + 33% + 33% = 99% — ah! That’s why there’s a note: “99% b/c of rounding error but total is 100%”

Because 180/534≈33.7%, 179/534≈33.5%, 175/534≈32.8% — when rounded to whole numbers, they become 34%, 34%, 33%? But they used 33% for all — probably rounded down.

Anyway, we’ll go with the completed tables as shown.

Now answer the questions:

A. Tables are already completed above.

B. How many students need swimming lessons?
→ Those who “Cannot Swim” = 74

C. How many students are there in 8th grade?
→ Grade 8 total = 179

D. How many of the 7th grade students cannot swim?
→ Grade 7, Cannot Swim = 60

E. How many students in grade 7 and 8 can swim?
→ Grade 7 Can Swim + Grade 8 Can Swim = 120 + 168 = 288

F. How many students are there altogether in grades 7, 8, and 9?
→ Grand total = 534

G. What is the relative frequency of students who are in 8th grade and cannot swim?
→ From right table: Grade 8, Cannot Swim = 2% (which is 11/534 ≈ 2.06% → rounded to 2%)

H. What percentage of 9th grade students can swim?
→ This is asking: Of 9th graders only, what % can swim?
→ Grade 9 total = 175, Can Swim = 172 → 172/175 ≈ 0.9829 → 98%?
Wait — but the handwritten answer says 32%. That’s because they’re using the grand total percentage again.

Looking back at the question: “What percentage of 9th grade students can swim?” — this could mean two things:

1. Percentage of ALL students who are 9th graders AND can swim → 172/534 ≈ 32% → matches handwritten answer.

2. Percentage OF 9th graders who can swim → 172/175 ≈ 98%

But since the context is the relative frequency table (which uses grand total), and the handwritten answer is 32%, we’ll go with 32% — meaning percent of total students.

However, strictly speaking, “percentage of 9th grade students” usually means out of 9th graders. But given the worksheet’s pattern, it’s likely expecting the grand total percentage.

To avoid confusion — let’s see the next question:

I. What percentage of students cannot swim?
→ Total cannot swim = 74 → 74/534 ≈ 13.86% → rounded to 14% ✔ matches handwritten.

J. What percentage of students are 9th graders?
→ 175/534 ≈ 32.77% → rounded to 33% ✔

So for H, even though wording suggests “of 9th graders”, the expected answer is 32% (as percent of total students). We’ll follow the worksheet’s logic.

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Problem 2: Glasses

Table:

| | Always wears glasses | Sometimes wears glasses | Never wears glasses | Total |
|---------|----------------------|--------------------------|---------------------|-------|
| Boys | 40 | ? | 161 | ? |
| Girls | 36 | 55 | 144 | ? |
| Total | ? | ? | ? | 484 |

Given: Total students = 484

Fill in missing values:

Boys sometimes wear glasses: Let’s call it X.

Girls total = 36 + 55 + 144 = 235

Boys total = 484 - 235 = 249

Then, Boys sometimes = Boys total - Always - Never = 249 - 40 - 161 = 48

Now, column totals:

Always: 40 + 36 = 76

Sometimes: 48 + 55 = 103

Never: 161 + 144 = 305

Check: 76 + 103 + 305 = 484 ✔

Now answer questions:

A. Complete the table — done above.

B. How many boys sometimes wear glasses? → 48

C. How many students wear glasses some of the time? → “Sometimes wears glasses” total = 103

D. How many students never wear glasses? → 305

E. Are there more boys or girls in the school?
→ Boys = 249, Girls = 235 → Boys > Girls by 14 → Boys (14 more)

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Final Answer:

Problem 1:

A. Tables completed as above.
B. 74
C. 179
D. 60
E. 288
F. 534
G. 2%
H. 32%
I. 14%
J. 33%

Problem 2:

A. Table completed: Boys sometimes = 48; Boys total = 249; Girls total = 235; Column totals: Always=76, Sometimes=103, Never=305
B. 48
C. 103
D. 305
E. Boys (14 more)