Math 7 ACC ✦ The Receipts

Answer Key

The walkthrough you came for, with explanations, math, and the occasional roast. No judgment if you peeked early — but reading the explanations is where the learning sneaks in.

Step-by-step With actual math No tears, hopefully
01 Stem-and-Leaf · Mean

"Why are stem-and-leaf plots called that? Because 'sideways number garden' was already taken."

Answer A — Boys: 26, Girls: 22

How we got there

Step 1: Decode the plot. The stem is the tens digit. Leaves on the LEFT belong to Boys (read right-to-left, toward the stem). Leaves on the RIGHT belong to Girls (read left-to-right). Yes, you have to read it like manga. Half of it, anyway.

Step 2: List the values.

Boys: 15, 18, 24, 26, 27, 32, 39   (7 values)
Girls: 10, 12, 15, 21, 23, 35, 38   (7 values)

Step 3: Mean = sum ÷ count.

Boys: (15+18+24+26+27+32+39) ÷ 7 = 181 ÷ 7 ≈ 25.86 → 26
Girls: (10+12+15+21+23+35+38) ÷ 7 = 154 ÷ 7 = 22
The boys did 4 more push-ups on average. Probably because they wouldn't stop talking about it.
02 Histogram · Reading bars

"A histogram is just a bar chart that went to college."

Answer B — The most common range was 3–5 hours per week.

How we got there

Counts from the bars:

0–2 hrs: 5 students  |  3–5 hrs: 14 students ← tallest
6–8 hrs: 7  |  9–11 hrs: 3  |  12+ hrs: 1
Total: 30 students

The 3–5 hour bar towers above the others, so B is correct.

Why the others crash

  • A: 7 + 3 + 1 = 11 students watched 6+ hours. That's not "more than half" of 30.
  • C: The bar covers 3–5 hours as a range. We don't know exactly how many watched 4 hours specifically. Histograms hide that detail on purpose.
  • D: 6–8 hrs = 7. 9+ hrs = 3 + 1 = 4. Not equal.
03 Population vs. Sample

"Tasting 8 cupcakes for 'quality control' is the dream job we should all aspire to."

Answer 200 cupcakes = POPULATION  ·  8 cupcakes = SAMPLE

How we got there

Population is the WHOLE group you care about. Sample is the smaller piece you actually measure to make a guess about the whole.

The baker cares about all 200 cupcakes. She just doesn't want to eat 200 cupcakes. (Probably. Possibly.) So she eats 8 and uses those to decide.

  • 200 cupcakes = the whole group → population
  • 8 cupcakes = the smaller group she actually tasted → sample
Quick memory hack: POPulation = the POP-ulous (large) group. Sample = the SMALL-ple. Yes, that's not a real word. Yes, you'll remember it forever now.
04 Random Sampling · Multi-select

"Surveying your 30 closest friends is research the same way eating cookie dough is baking."

Answers C and E

How we got there

A random sample means EVERY single person in the population has an EQUAL chance of being chosen. Not "everyone who walks past the cafeteria." Not "everyone with a phone." Everyone.

  • ✅ C — Bingo machine: Every student numbered, every number equally likely. Random!
  • ✅ E — Random number generator on full roster: Same idea, fancier tech. Random!

The not-random crew

  • A: First 30 in the cafeteria? You miss everyone who eats at home or doesn't eat lunch. Biased.
  • B: Football + cheer = athletes. Excludes the rest of the school. Biased.
  • D: Social media replies = people who saw the post AND chose to respond. Self-selection bias.
  • F: Your friends are not a random slice of the school. They're a random slice of your taste in friends.
05 Proportions

"If 3 out of 5 prefer pizza, the other 2 are wrong but we still respect them."

Answer B — 2,700 students

How we got there

"3 out of 5" means 3/5 of the surveyed students prefer pizza. We just need 3/5 of 4,500.

Method 1 (fraction): 3/5 × 4,500 = (3 × 4,500) ÷ 5 = 13,500 ÷ 5 = 2,700
Method 2 (decimal): 3 ÷ 5 = 0.6, then 0.6 × 4,500 = 2,700
Method 3 (chunks): 1/5 of 4,500 = 900. So 3/5 = 900 × 3 = 2,700

Three different ways. Use whichever your brain likes best — they all land in the same place.

06 Identifying the Population

"Your soccer team is a sample. The whole country is a vibe."

Answer C — All middle school soccer players in the country

How we got there

Population = the group the conclusion is supposed to apply to. Read the question carefully: who does the league want to learn about?

"...estimate the average number of goals scored by all middle school soccer players in the country."

Boom. They want to know about ALL middle school soccer players. That's the population. Your soccer team is just the (probably-not-very-good) sample they chose to estimate from.

A common trap: people pick "your soccer team" because that's the group that was surveyed. But that's the SAMPLE. The population is what you're trying to draw conclusions about.
07 Dot Plot · Comparing data sets

"Two data sets walk into a bar... wait, no, a number line. Sorry."

Answer A — The means are about 25 points apart

How we got there

Step 1: Read the dots.

Set 1 (●): 10, 20, 30, 30, 30, 40, 40, 50   (8 values)
Set 2 (✕): 30, 50, 50, 60, 60, 60, 60, 80   (8 values)

Step 2: Compute means.

Set 1: 250 ÷ 8 = 31.25
Set 2: 450 ÷ 8 = 56.25
Difference: 56.25 − 31.25 = 25 ✓

Why the others miss

  • B Medians: Set 1 median = 30, Set 2 median = 60. They're 30 apart, not 10.
  • C Modes: Set 1 mode = 30 (appears 3×). Set 2 mode = 60 (appears 4×). Neither is 40.
  • D Ranges: Set 1 range = 50 − 10 = 40. Set 2 range = 80 − 30 = 50. Not both 50.
08 Building a Stem-and-Leaf Plot

"Leaves grow outward from the stem. Like in nature. Stay with me here."

Answer A

How we got there

Two rules to live by:

  1. Leaves are written so they get LARGER as you move away from the stem.
  2. Set A goes left of stem, Set B goes right of stem (the question's labels tell you which side is which).

The data sorted:

Set A: 23, 27, 35, 41, 48, 52
Set B: 19, 25, 33, 36, 47, 54

Why each wrong answer is wrong:

The imposters

  • B: Set A and Set B leaves are on the wrong sides. Read it carefully — the 9 ends up next to stem 1 on the LEFT, but 19 is in Set B (right side). Swap fail.
  • C: Look at stem 2 row: it shows "3, 7" on the left. But on the LEFT side, leaves should grow away from the stem, so 23 and 27 → "7, 3" (reading toward the stem gives 3, 7). Same problem at stem 4.
  • D: Stem 3 right side shows "6, 3" — but Set B values are 33 and 36, so right side should be "3, 6" (smallest near stem). Reverse fail.
09 Sample Bias · Multi-select

"Surveying gym-goers about TV shows is like asking lifeguards what they think of swimming. They're enthusiasts."

Answers B, C, and E

How we got there

Maya's sample is everyone leaving the gym Saturday morning. Let's evaluate each statement:

  • ✅ B — Excludes non-gym teens: Yep. If you skipped the gym, Maya literally cannot survey you. Whole demographics missed.
  • ✅ C — Biased toward people who exercise: 100% of her sample exercises. That's the definition of bias.
  • ✅ E — NOT a random sample: Random means every teen has an equal chance. Couch-dwelling teens have zero chance here. Not random.

The false statements

  • A: Sample includes the entire population? No, that would be a census. Maya only got 20 people, not all teens.
  • D: Representative? It's basically the opposite. Gym-goers ≠ all teens.
  • F: Random sample? See E. They cannot both be true. (Math is logical like that.)
Maya should number all the teens in town and pick from a random number generator. Or, you know, just send a survey to every school. Wild idea.
10 Interquartile Range (IQR)

"IQR: the part of statistics that finally rewards you for putting things in order."

Answer A — Week 1 IQR = 16, Week 2 IQR = 14

How we got there

The 4-step IQR routine:

  1. Sort the data smallest to largest.
  2. Find the median (the middle number) — this splits the data in half.
  3. Find Q1 = median of the lower half. Find Q3 = median of the upper half.
  4. IQR = Q3 − Q1.

Week 1: 22, 35, 18, 41, 29, 38, 25

Sorted: 18,  22,  25,  29,  35,  38,  41
Median = 29 (the 4th value). Lower half: 18, 22, 25 → Q1 = 22.
Upper half: 35, 38, 41 → Q3 = 38.
IQR = 38 − 22 = 16

Week 2: 30, 42, 28, 35, 19, 51, 33

Sorted: 19,  28,  30,  33,  35,  42,  51
Median = 33. Lower half: 19, 28, 30 → Q1 = 28.
Upper half: 35, 42, 51 → Q3 = 42.
IQR = 42 − 28 = 14
The IQR is the "middle 50%" — it ignores the extreme high and low values, which is why it's a more reliable measure of spread than range. Range can be ruined by one weird outlier (like the day Mr. Lee accidentally watched 3 hours of cat videos).

Quick Answer Key

For when you just need to check your bubbles and move on with your day.

Q1A
Q2B
Q3POP / SAM
Q4C, E
Q5B
Q6C
Q7A
Q8A
Q9B, C, E
Q10A