Drawing Inferences from Random Samples

Grade 7 Statistics — 7.SP.A.2

In this lesson:

• Use sample data to estimate population characteristics
• Understand why different samples give different estimates
• Evaluate the reliability of statistical claims

What You Will Learn Today

1. Estimate a population characteristic from a random sample
2. Express estimates as proportions, percents, and counts
3. Explain why different samples give different estimates
4. Use multiple samples to gauge plausible estimate ranges
5. Describe how sample size affects reliability
6. Evaluate estimates based on the sampling process

Can 40 Students Speak for 800?

You already know:

• A random sample tends to be representative of the population
• Sampling from a class roster gives everyone an equal chance

Today's question: If 26 out of 40 randomly selected students have a pet, what can we say about all 800 students in the school?

Statistical Inference: Sample to Population

Statistical inference means using data from a sample to draw a conclusion about a population.

• The sample must be random for the inference to be valid
• The conclusion is an estimate — not a guaranteed fact
• Language matters: say "we estimate" — never "we know for sure"

From Sample to Population: Three Steps

Three steps move from sample data to a population estimate:

• Step 1: Find the proportion in the sample
• Step 2: Apply that proportion to the population (as a percent)
• Step 3: Estimate the count in the population

Pet Ownership: Applying the Three Steps

26 of 40 randomly surveyed students have a pet. School has 800 students.

Step 1:

Step 2: We estimate about 65% of all 800 students have a pet

Step 3: students estimated to have a pet

Quality Control: Three Steps in Action

Factory tests 100 random widgets; 4 defective. Produces 10,000/day.

Step 1:

Step 2: We estimate about 4% of all widgets are defective

Step 3: estimated defective per day

Your Turn: The Voter Poll Problem

A pollster randomly calls 200 of a city's 5,000 voters; 130 prefer Candidate A.

Use the three-step framework:

1. What is the sample proportion?
2. What percent likely prefer Candidate A?
3. Estimate how many of the 5,000 voters prefer Candidate A

Work all three steps before the next slide

Check: What Does the Inference Tell Us?

We estimate about voters prefer Candidate A

Is 3,250 exact? No — it is an estimate. The true count is likely close, but not necessarily 3,250.

Always say "we estimate about 65%" — never drop the hedge

What If We Tried Again?

We surveyed 40 students and estimated 65% have pets.

What if we surveyed a different group of 40 random students?

• Would we get exactly 65% again?
• Would we get a different number?
• How different might it be?

These questions are the heart of what we explore next...

Sampling Variability: Normal, Not an Error

Sampling variability: different random samples produce different estimates.

• This is expected — not a sign something went wrong
• Each random sample draws a slightly different mix of people
• Estimates tend to cluster near the true population value

The Tile Simulation in Action

A bag holds 100 tiles — true proportion unknown.

10 groups each draw 10 tiles:

• Group 1: 6 red → 60%
• Group 2: 4 red → 40%
• Group 3: 7 red → 70%
• Same bag, different estimates each time

Dot Plot of Sample Estimates

When many groups sample from the same population, their estimates spread around the true value.

How to Read a Dot Plot

In a dot plot of sample estimates, look for:

• Center: Average — best guess for the true proportion
• Spread: Distance between highest and lowest estimates
• Cluster: Where most estimates fall — plausible range

The mean of all estimates beats any single sample estimate.

Quick Check: Why Different Estimates?

Two groups each drew 10 tiles from the same bag.

• Group A: 7 red → estimated 70%
• Group B: 4 red → estimated 40%

Why did they get different estimates from the same bag?

Think before the next slide...

Guided Practice: Analyze Sample Proportions

8 random samples from the same population:

0.50, 0.60, 0.40, 0.70, 0.50, 0.60, 0.80, 0.40

Tasks:

1. Create a dot plot of these estimates
2. Find the range (highest − lowest)
3. Calculate the mean of all estimates
4. Write an inference statement for the population proportion

Dot Plot Practice: Check Your Answers

Data ordered: 0.40, 0.40, 0.50, 0.50, 0.60, 0.60, 0.70, 0.80

"We estimate approximately 56% of the population is red."

Does the Sample Size Matter?

Samples of 10 produce variable estimates.

What if we drew samples of 20 instead?

• Would estimates still vary?
• Would they vary as much?
• Which size gives more reliable estimates?

Let's compare...

Larger Samples Produce Less Variability

When sample size increases:

• Estimates cluster more tightly around the true value
• The range of estimates decreases
• Any single estimate becomes more reliable

Opinion polls use 1,000+ respondents — not because the population is huge, but for reliability.

Comparison: Small vs. Large Samples

Quick Check: Which Is Which?

Two dot plots from repeated sampling of the same population:

• Plot A: Estimates range from 0.30 to 0.90
• Plot B: Estimates range from 0.50 to 0.70

Which plot came from the larger sample size? How do you know?

Write one sentence before moving on

Worked Example: How Precise Do We Need?

A factory needs the defect rate within 2%. Test 20 or 200 widgets?

• 20 widgets: estimates vary by 10+ percentage points
• 200 widgets: estimates cluster much more tightly

Answer: Test 200 — larger sample meets the 2% precision requirement.

Connecting Inference to Real-World Claims

Is a statistical estimate trustworthy? Three conditions determine it:

1. Was the sample random?
2. Is the sample size adequate?
3. Is the population clearly defined?

All three must pass for the inference to be reliable.

Three Conditions for Trustworthy Inference

Every reliable inference must satisfy three conditions:

• ✓ Random: Every member had a chance of being selected
• ✓ Adequate size: Large enough to reduce variability
• ✓ Defined population: Exactly who the population is is clear

Library Books: Evaluating the Inference

Librarian randomly samples 50 of 2,000 books; 12 need repairs → estimates ~480 need repairs.

High trust — all three conditions pass.

Hallway Survey: Why This Fails

6 of 8 hallway students prefer longer lunch. Principal: "75% of students agree."

Low trust — two conditions fail.

Guided Practice: Evaluate the Researcher's Claim

Researcher randomly selected 500 seventh-graders from 10 states; 68% prefer group projects.

Evaluate using the three conditions:

1. Is the sample random?
2. Is 500 adequate?
3. Is "all seventh-graders" defined?

Rate: high / medium / low — explain in one sentence

Practice: Evaluate the Blogger's Claim

A blogger asked 12 regular readers: coffee or tea? 8 said coffee. She claims most people prefer coffee.

Complete the checklist:

Write trust level and one sentence of justification

Practice: Answers to the Blogger Claim

Low trust. All three fail. Random + adequate size + defined population are all required.

Key Ideas: Sample, Estimate, and Variability

✓ Inference flows from sample to population — proportion → percent → count

✓ Always hedge: "We estimate approximately..." — never claim a population fact

✓ Variability is normal — different samples give different estimates; this is expected

✓ Mean of multiple samples beats any single estimate for reliability

Common Mistakes to Watch Out For

Skip the hedge — always say "we estimate"

Variability = error — different samples differ; normal

Bigger pop → bigger — reliability depends on , not

Any sample or one sample works — only large random samples give reliable inferences

What's Next: Comparing Two Populations

Now you can make inferences from a single population.

Coming up — 7.SP.B:

• Collect data from two different populations
• Compare the distributions using visual displays
• Draw inferences about differences between groups

Example: Do 7th graders and 8th graders have different screen time habits?