4. Introduction to Confidence Intervals

b. Interpretation and Width Factors

Yurk

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:

  1. What does this interval mean?
  2. 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.

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:

  1. Confidence level — we just saw this (higher → wider)
  2. Sample size n — how much data do we have?
  3. 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.

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.

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.

  1. How big is each bootstrap sample? n = 200 — same size as the original sample.

  2. 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).

  3. To get the 95% CI, which percentiles do you read off? The 2.5th and 97.5th — chopping 2.5% off 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?

  1. “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.

  2. “We’re 95% confident that p is between 0.24 and 0.36.” Correct — the standard phrasing.

  3. “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.”

Hypothesis Test or CI?

  1. 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).

  2. 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.

  3. 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?

  1. Same sample data — but switch from a 95% CI to a 99% CI. Wider — more confidence requires more cushion on each side.

  2. Same \hat{p} — but a larger sample: n goes from 100 to 500. Narrower — more data means less variability in \hat{p}.

  3. 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

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