Math 115: Introduction to Statistics – 4. Introduction to Confidence Intervals

Recall: Where We Left Off

For Candidate X, \(\hat{p} = 0.7\) from a sample of 30.

Bootstrap 95% CI: (0.533, 0.834)

Today:

What does this interval mean?
What shapes its width?

Interpreting Confidence Intervals

Our 95% CI for Candidate X: (0.533, 0.834)

Correct interpretation:

“We are 95% confident that the true proportion of voters who support Candidate X is between 0.533 and 0.834.”

What this means:

Values inside the interval are plausible for \(p\)
Values outside the interval are implausible for \(p\)
The CI contains the observed \(\hat{p} = 0.7\) near the center

What Does “95% Confident” Mean?

Common Misinterpretation

❌ WRONG: “There is a 95% probability that \(p\) is in this interval”

✅ CORRECT: “We are 95% confident that this interval contains \(p\)”

Why the distinction?

The parameter \(p\) is fixed (but unknown)
It either is or isn’t in our interval
The 95% refers to our confidence in the method, not probability about this specific interval
If we repeated this process many times using new samples of 30, about 95% of intervals would contain \(p\)

A Bigger Poll

Candidate X’s campaign commissions a follow-up poll — this time 400 voters.

220 say yes → \(\hat{p} = 0.55\)
More data, finer percentile resolution

We’ll use this poll for the rest of the lecture as we explore what shapes the width of a CI.

Different Confidence Levels

So far we’ve used 95% confidence — but the choice is up to us.

Generalized percentile method. For an X% confidence interval, we take the middle X% of the bootstrap distribution. That leaves \(\frac{100-X}{2}\%\) in each tail.

Confidence Level	Each Tail	Percentiles
90%	5%	5th to 95th
95%	2.5%	2.5th to 97.5th
99%	0.5%	0.5th to 99.5th

95% is the most common choice — like the 0.05 cutoff for hypothesis tests, it’s a convention, not a special value.

Different Levels on the Follow-Up Poll

Same follow-up poll bootstrap distribution; watch the bounds shift as we change the level.

Estimate the CI Bounds

The next slide shows the bootstrap distribution for the follow-up poll (1000 resamples).

Estimate the 95% and 90% CI bounds by reading bar heights from each tail.

Hint: a 95% CI excludes about 2.5% × 1000 = 25 resamples from each tail. A 90% CI excludes 5% × 1000 = 50. Advance for the answer.

Reading the Bounds (Your Turn)

Counts above each bar — try it before advancing.

Let’s work through this together.

Take a couple of minutes here. Add up the bin counts from the left tail until you reach about 25 — that’s where the 95 percent lower bound lands. Same on the right. Then keep going on each side until you hit about 50 — those are the 90 percent bounds. Write down all four estimates before clicking forward. Now let’s check. The orange bars are the ones entirely outside the 95 percent bounds. They sum to less than 25 on each side, so the bound has to slide partway into the next bar to pick up the remaining resamples. There are the 95 percent bounds — the dashed red lines. Each one cuts partway through the next bar in from the tail. For 90 percent confidence we want 50 in each tail instead of 25, so we extend further. The purple bars now reach deeper into the distribution, but again come up just shy of 50. And there are the 90 percent bounds — the purple dotted lines. Notice they sit inside the 95 percent bounds: same data, lower confidence, narrower interval.

Why More Confidence Means More Width

Think back to the darts and bullseye picture.

95% confidence: draw a circle around our dart big enough that the bullseye is inside 95% of the time across many possible darts.
99% confidence: to capture the bullseye 99% of the time, the circle has to be bigger.

The more often we want our interval to contain \(p\), the wider it has to be.

Trade-off: confidence vs. precision.

What Affects CI Width?

Three factors shape how wide a CI ends up being:

Confidence level — we just saw this (higher → wider)
Sample size \(n\) — how much data do we have?
The value of p̂ — proportions near 0.5 vary more than those near 0 or 1

Let’s look at the other two.

Width Factor — Sample Size

We’ll hold \(\hat{p} = 0.60\) and the confidence level fixed and vary \(n\). Watch as the 95% CI shrinks as the sample gets larger.

The setup for the next four frames is the same hypothetical survey at four different sample sizes — 20, then 50, then 100, then 500 — all landing on p hat equal to 60 percent. Watch what happens to the 95 percent confidence interval as the sample grows. The first frame is n equals 20: the bootstrap distribution is wide, and the confidence interval runs from 40 percent up to 80 percent — a 40-point range that pins p down only loosely. At n equals 50, the confidence interval narrows to roughly 46 to 74 percent — a 28-point range. At n equals 100, narrower still — about 50 to 69 percent, a 19-point range. And at n equals 500, the bootstrap distribution has tightened sharply: the confidence interval runs from about 56 to 64 percent, just an 8-point range. Each step up in sample size produces a noticeably narrower interval.

Sample Size: The Lever We Control

Larger sample → narrower CI. As we collect more data, each individual observation matters less to \(\hat{p}\) — a sample of 1000 barely budges when one voter changes their mind, while a sample of 30 swings noticeably. With a bigger sample, and more reliable \(\hat{p}\), the range of plausible values for \(p\) shrinks around it.

At a given confidence level, a narrower CI tells us more about \(p\) — “\(p\) is between 0.55 and 0.65” is much sharper than “between 0.30 and 0.90.”

That’s why \(n\) is the lever we pull in study design — it’s the only way to narrow the CI without trading something else away. Confidence level is also up to us, but tightening it costs confidence.

The animation we just walked through showed what changing n does to the interval; this slide is the why behind it. As n grows, each individual observation matters less to p hat — a sample of one thousand barely budges when one voter changes their mind, while a sample of thirty swings noticeably. The more stable p hat is, the tighter the range of plausible values around it — and a tighter range tells us more, since p between zero point five five and zero point six five is genuinely sharper than p between zero point three and zero point nine. That’s why n earns its lever label: confidence level is also up to us but trades against precision, and p hat is something we observe rather than choose, so sample size is the one knob we can turn that doesn’t sacrifice anything statistical — though it does cost dollars, time, and recruitment effort.

Width Factor: The Value of p̂

For a proportion, the spread of the bootstrap distribution depends on \(\hat{p}\) itself:

\(\hat{p}\) near 0.5 → maximally variable → widest CI
\(\hat{p}\) near 0 or 1 → minimally variable → narrowest CI

Intuition: if 99% of voters support the candidate, almost every bootstrap sample lands near 0.99 — there’s little room for variability. If 50% do, bootstrap samples can swing more in either direction.

Same n, Different p̂

Hold \(n = 100\) fixed; vary \(\hat{p}\).

Same \(n\) — but the variability (and the CI) is widest near \(\hat{p} = 0.5\).

Connecting to Hypothesis Testing

Back to the medical consultant from topic 3: 3 complications in 62 surgeries, \(\hat{p} = 0.048\).

The national rate of \(0.10\) sits inside the 95% CI \((0, 0.113)\).

Same Conclusion, Two Methods

Hypothesis test (topic 3): failed to reject \(H_0: p = 0.10\) (p-value \(\approx 0.111\))
Confidence interval: \(0.10\) is in the plausible range — about \((0, 0.11)\)

Two methods, same verdict — but the CI does more:

A hypothesis test answers a yes-or-no question about one specific value of \(p\).
A CI gives the whole range of plausible \(p\) values, so you can read off the verdict for any candidate at a glance.

Bootstrap Recipe Check

You sample \(n = 200\) Hope students; 60 own a car, so \(\hat{p} = 0.30.\) Answer each before clicking.

How big is each bootstrap sample? \(n = 200\) — same size as the original sample.
Where is the bootstrap distribution centered? Around \(\hat{p} = 0.30\) — bootstraps re-mix the observed sample, so they cluster near the observed value (not near some hypothesized \(p\)).
To get the 95% CI, which percentiles do you read off? The 2.5th and 97.5th — chopping 2.5% off each tail.

Over the next several slides we’ll run through all of the major confidence-interval ideas from this lecture as a series of your-turns — bootstrap mechanics, what a confidence interval actually means, when to use one versus a hypothesis test, and how width responds to changes in n, p hat, and confidence level. Each slide poses its questions first; pause and work through them before advancing for the answers. To start, the bootstrap recipe: suppose we sample 200 Hope students; 60 own a car, so p hat is 30 percent. Three questions about the bootstrap recipe for this sample — take a moment with each before clicking. How big is each bootstrap sample? Same size as the original — n equals 200, sampled with replacement. Where is the bootstrap distribution centered? At p hat — the value we observed — not at some hypothesized p sub zero. That’s a key contrast with the null distribution from hypothesis testing. Which percentiles for a 95 percent confidence interval? The 2.5 and 97.5 — chopping 2.5 percent from each tail.

Interpret This CI

A 95% bootstrap CI for the proportion of Hope students who own a car is (0.24, 0.36). Correct or wrong?

“There’s a 95% probability that the true proportion is in (0.24, 0.36).” Wrong — \(p\) is fixed; the 95% is about our method, not this interval.
“We’re 95% confident that \(p\) is between 0.24 and 0.36.” Correct — the standard phrasing.
“If we ran this study many times, about 95% of the resulting CIs would contain the true \(p\).” Correct — the long-run-frequency view of “95% confident.”

Three interpretation candidates for the given confidence interval. For each, decide if it’s a correct interpretation before clicking. First statement — claiming there’s a 95 percent probability that p is in the interval. Wrong. p is a fixed number — not random — so probability statements about where it sits don’t apply. The 95 percent describes our method, not this interval. Second statement — the standard “95 percent confident” phrasing. Correct. We’re confident in the method that produced this interval, and we apply that confidence to the parameter. Third — the long-run-frequency framing: repeat the study many times, 95 percent of the resulting confidence intervals cover the true p. Also correct. That’s the rigorous interpretation behind “95 percent confident.”

Hypothesis Test or CI?

H-test or CI? “Is the proportion of students dissatisfied with the cafeteria greater than 30%?” Hypothesis test — testing whether \(p\) exceeds a specific value (\(p_0 = 0.30\)).
H-test or CI? “What range of values are plausible for the proportion of dissatisfied students?” Confidence interval — estimating \(p\) with no specific value in mind.
Where is the simulated distribution centered in each case? HT: at the null value \(p_0\) (assumed true). CI: at \(\hat{p}\) (the observed value).

Predict the CI Change

Each scenario changes one thing. Does the CI get wider, narrower, or stay the same?

Same sample data — but switch from a 95% CI to a 99% CI. Wider — more confidence requires more cushion on each side.
Same \(\hat{p}\) — but a larger sample: \(n\) goes from 100 to 500. Narrower — more data means less variability in \(\hat{p}\).
Same \(n\) — but \(\hat{p}\) moves from 0.50 to 0.90. Narrower — proportions near 0 or 1 vary less than proportions near 0.5.

References

Introduction to Modern Statistics (2e) textbook by Mine Cetinkaya-Rundel and Johanna Hardin
Chapter 12: Confidence Intervals with Bootstrapping