Point Estimate and Simulating Its Variability

Lesson 1 of 2: From One Number to a Distribution

In this lesson:

• Use a sample statistic as a point estimate of a parameter
• Simulate how much that estimate varies across samples

Learning Objectives for This Unit

By the end of this two-lesson unit, you should be able to:

1. Use a statistic as a point estimate
2. Simulate an estimate's sampling distribution
3. Estimate a margin of error from spread
4. Report and interpret an interval estimate
5. Explain how sample size affects the margin

"52%, Plus or Minus 3" — Where From?

A poll reports 52% support, ± 3 points.

• Everyone reads the 52 and ignores the rest
• But the ± 3 is the real content — how uncertain we are

Where does that ± come from? Not a guess, not measurement error. Let's find out.

Recall: The Estimate Already Bounces

• Three random samples → three estimates (from A.1)
• We know it bounces — but never measured how much

The Point Estimate: Best Single Guess

A sample statistic is a point estimate of the parameter.

• estimates ; estimates
• Our sample of 60 gave — the point estimate of

It's the best single guess — but says nothing about its own precision.

The Mean Context: 27 Minutes

A random sample of 80 workers had a mean commute of 27 minutes.

• Point estimate of the population mean: minutes
• Same move as the proportion — best single guess for

Proportion or mean, the method is identical. We'll carry both.

A Bare Estimate Overstates Certainty

Another random sample would land elsewhere — 0.62, 0.68, ...

• Reporting just "0.65" makes it sound exact
• We really only know it's in a neighborhood

An honest estimate gives a value and its precision. So: how far off could 0.65 be?

Give the Estimate; Why Incomplete?

A biologist nets 50 fish, mean length 22 cm.

• What is the point estimate of the true mean length?
• Why is that single number incomplete?

Commit both. Think: what if she netted a different 50 fish?

We Know It Bounces — Now Measure It

To measure the bounce: draw many same-size samples, record each estimate.

• The spread of those estimates = how much ours might be off
• That collection is the sampling distribution

Same simulation idea as before, aimed at a new question. Let's build it.

The Problem: What Do We Sample From?

To simulate many samples, we need a population to draw from.

• But the true proportion is unknown — that's what we're estimating!
• We can't sample from a population we can't see

So what do we sample from? Sit with this — there's a clever resolution.

Resampling: The Sample Stands In

• The observed sample is our best stand-in for the population
• Draw new samples from it, with replacement (resampling)

One Resampling Trial, Same Size n = 60

Each resample must be the same size as the real sample — 60.

• The question is how much a sample of this size bounces
• A different size would give the wrong precision

Same rule as before: one trial matches the real data collection.

Repeat the Trial Hundreds of Times

• Resample 60 → compute
• Resample 60 → compute another
• ... hundreds of times, recording every estimate

Each is a value our study might have produced. Hundreds reveal a stable shape.

The Sampling Distribution of the Estimate

• Centers near our estimate, 0.65
• Spreads from about 0.53 to 0.77 — the bounce

Center Confirms; Spread Is the Output

• Center ≈ 0.65 → confirms the estimate is reasonable (not new)
• Spread → how much the estimate varies — the precision

We extract the spread, not the center. Next lesson, the spread becomes the margin.

Same Machine as Before, New Question

This is the same simulation machine as model-fitting.

• Before: "is this result surprising under a model?"
• Now: "how much does my estimate bounce?"

Same tool, different question — a recurring theme in statistics.

Describe the Resampling Trial Yourself

A survey of 200 voters finds 45% favor a measure.

• Describe the resampling trial (sample from? size? record?)
• What do the distribution's center and spread represent?

Write all of it before advancing. Describing the trial clearly is the skill.

Key Takeaways From Lesson One

✓ A statistic is a point estimate — incomplete alone
✓ Resample same-size samples to build the distribution
✓ The center confirms; the spread is the output

0.65 is the middle of a range, not the answer
The hat in means estimate of

Coming Up Next: Making the Number

In Lesson 2, you'll turn the spread into the margin of error — about two standard deviations.

That makes "0.65" into "0.65 ± 0.12": the honest range every poll reports.