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Estimation: Confidence Intervals as Ranges of Plausible Values

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From a Single Guess to a Range of Plausible Values

When you use a sample to learn about a population, you usually want to estimate an unknown population value (a parameter) such as a population mean or a population proportion. A point estimate is a single best guess from your sample (for example, the sample mean or sample proportion). A confidence interval adds context: it gives a range of plausible values for the parameter, based on how much sampling variation you expect.

Think of it this way: a point estimate answers “What is my best guess?” A confidence interval answers “How wrong could my guess reasonably be, given the sample size and variability?”

Point estimates you will use most often

Population mean (unknown): estimated by the sample mean x̄.
Population proportion (unknown): estimated by the sample proportion p̂.

What a Confidence Interval Is (and Is Not)

What it is

A confidence interval (CI) is constructed from your data using a rule that is designed to capture the true parameter a certain percentage of the time in the long run.

A generic CI has the form:

estimate ± (critical value) × (standard error)

Where:

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estimate is x̄ or p̂.
standard error (SE) measures typical sampling variation of the estimate.
critical value depends on the chosen confidence level (e.g., 90%, 95%, 99%).

What it is not

Not “a 95% probability the parameter is in this interval” for your one fixed dataset. After you compute the interval, the parameter is either in it or not.
Not a guarantee that the interval contains the truth.
Not a statement about individual outcomes (it is about a population parameter).

The Long-Run Interpretation: Many Intervals from Repeated Samples

The most reliable way to understand confidence is to imagine repeating the same sampling process many times, each time building a CI using the same method.

Suppose you build 100 separate 95% confidence intervals from 100 independent samples of the same size from the same population:

About 95 of those intervals will contain the true parameter.
About 5 will miss it (sometimes by a little, sometimes by a lot).

Visually, you can picture a “stack” of intervals. Most cross the true value (a vertical reference line), and a few do not. The confidence level is about the procedure’s hit rate, not about uncertainty in a single finished interval.

Why Confidence Intervals Change: Sample Size, Variability, Confidence Level

1) Sample size: larger samples give narrower intervals

Standard errors shrink as sample size grows, because averaging (or aggregating) more information reduces random fluctuation.

For a mean, the SE is roughly proportional to 1/√n.
For a proportion, the SE is also proportional to 1/√n (with an additional dependence on p).

Practical implication: to cut the margin of error in half, you typically need about four times the sample size.

2) Variability: more spread means wider intervals

If the underlying measurements vary a lot from person to person (or unit to unit), the sample mean will bounce around more from sample to sample. That larger bounce shows up as a larger SE, which widens the CI.

For proportions, variability is highest near 50% and lower near 0% or 100%. That’s why estimating a proportion near 0.5 tends to require larger samples to achieve the same margin of error.

3) Confidence level: more confidence means wider intervals

Higher confidence requires a larger critical value, which multiplies the SE and increases the margin of error.

A 99% CI is wider than a 95% CI from the same data.
A 90% CI is narrower than a 95% CI from the same data.

This is a trade-off: higher confidence means you cast a wider net.

Confidence Intervals for a Population Mean (Intuitive Level)

Goal: estimate an unknown population mean (average) using a sample mean x̄.

Core idea

Your sample mean is unlikely to equal the population mean exactly. A CI uses the estimated typical error of x̄ to create a plausible range for the true mean.

A common form is:

x̄ ± t* × (s / √n)

Where s is the sample standard deviation and t* is a critical value (often from a t distribution) chosen to match the confidence level.

Step-by-step: building a CI for a mean

Compute the point estimate: calculate x̄.
Measure sample variability: compute s.
Compute the standard error: SE = s/√n.
Choose a confidence level (e.g., 95%) and obtain the corresponding t* for your sample size.
Compute margin of error: ME = t* × SE.
Form the interval: (x̄ − ME, x̄ + ME).

Example (mean): average delivery time

A company samples n = 64 deliveries and finds an average delivery time of x̄ = 42.0 minutes with sample standard deviation s = 12.0 minutes. For an intuitive 95% CI, use a critical value near 2 (exact t* depends slightly on n).

SE = 12 / √64 = 12 / 8 = 1.5
ME ≈ 2 × 1.5 = 3.0
CI ≈ 42.0 ± 3.0 → (39.0, 45.0)

Interpretation: using this method, the data support a plausible range for the population mean delivery time of about 39 to 45 minutes.

Confidence Intervals for a Population Proportion (Intuitive Level)

Goal: estimate an unknown population proportion (a rate, percentage, or probability) using a sample proportion p̂.

Core idea

If you sample n individuals and observe a “success” in x of them, then p̂ = x/n. A CI accounts for the fact that p̂ varies from sample to sample.

A common approximate form is:

p̂ ± z* × √(p̂(1 − p̂)/n)

Where z* is a critical value from the standard normal distribution (about 1.64 for 90%, 1.96 for 95%, 2.58 for 99%).

Note: In practice, some methods perform better than the simple formula above, especially for small samples or proportions near 0 or 1. The meaning and interpretation of the CI remain the same.

Step-by-step: building a CI for a proportion

Count successes: record x out of n.
Compute the point estimate: p̂ = x/n.
Compute the standard error: SE = √(p̂(1 − p̂)/n).
Choose a confidence level and corresponding z*.
Compute margin of error: ME = z* × SE.
Form the interval: (p̂ − ME, p̂ + ME), usually reported as percentages.

Example (proportion): app conversion rate

An app team samples n = 400 visitors and observes x = 92 sign-ups. Then p̂ = 92/400 = 0.23. For a 95% CI, use z* ≈ 1.96.

SE = √(0.23 × 0.77 / 400) ≈ √(0.1771 / 400) ≈ √0.0004428 ≈ 0.0210
ME ≈ 1.96 × 0.0210 ≈ 0.041
CI ≈ 0.23 ± 0.041 → (0.189, 0.271)

Interpretation: the data support a plausible range of about 18.9% to 27.1% for the population conversion rate, using this 95% method.

Reading a Confidence Interval Like a Decision-Maker

Center: what value is most supported?

The interval is centered around the point estimate. If the interval is (39, 45), values near 42 are most consistent with the sample.

Width: how precise is the estimate?

Two studies can have the same point estimate but different precision. A narrow CI suggests high precision; a wide CI suggests substantial uncertainty.

Compare to a benchmark

Often you care whether the parameter is above/below a target:

Mean delivery time target: 40 minutes. If the CI is (39, 45), the data do not clearly support that the mean is below 40.
Conversion target: 25%. If the CI is (18.9%, 27.1%), the data are consistent with being below or above 25%.

Confidence Level vs. Margin of Error: A Concrete Comparison

Suppose you have the same estimate and SE, and you only change the confidence level:

Confidence level	Typical critical value	Effect on interval width
90%	`z* ≈ 1.64`	Narrower
95%	`z* ≈ 1.96`	Wider
99%	`z* ≈ 2.58`	Widest

If your SE is 0.021 (as in the conversion example), then:

90% ME ≈ 1.64 × 0.021 = 0.034
95% ME ≈ 1.96 × 0.021 = 0.041
99% ME ≈ 2.58 × 0.021 = 0.054

The data did not change; only your desired long-run coverage did.

Common Misinterpretations (and Better Alternatives)

Misinterpretation 1: “There is a 95% chance the true value is in my interval.”

Why it’s wrong: the parameter is fixed; your interval is the random object (it would change if you resampled).

Better: “This method produces intervals that contain the true value 95% of the time in repeated sampling.”

Misinterpretation 2: “A wider interval means the data are bad.”

Why it’s misleading: width reflects uncertainty, which can be appropriate when samples are small or variability is high.

Better: “The estimate is imprecise; to narrow it, increase sample size or reduce measurement noise.”

Misinterpretation 3: “If 0 is not in the interval, the effect is important.”

Why it’s wrong: excluding 0 (for differences) or excluding 1 (for ratios) relates to statistical detectability, not practical impact. With large samples, tiny effects can be statistically detectable.

Better: judge importance by the size of the effect and whether the CI lies within a practically meaningful range. For example, a conversion lift CI of (+0.2%, +0.6%) might be statistically clear but operationally minor.

Misinterpretation 4: “Overlapping confidence intervals mean there is no difference.”

Why it’s unreliable: overlap is not a definitive test of difference; it depends on how the intervals were built and whether samples are independent.

Better: if comparing groups is the goal, construct a CI for the difference (or use a dedicated comparison method) rather than comparing two separate CIs informally.

Misinterpretation 5: “The confidence level measures data quality.”

Why it’s wrong: confidence level is a choice you make; it controls long-run coverage by widening or narrowing the interval.

Better: treat confidence level as a policy decision about risk tolerance, and treat CI width as the indicator of precision.

Practical Checklist Before You Trust a Confidence Interval

Is the sample reasonably representative? If the sampling process is biased, a narrow CI can still be centered on the wrong value.
Is the sample size adequate for the method? Very small samples can make simple approximations unreliable, especially for proportions near 0 or 1.
Are observations reasonably independent? If data points are strongly related (e.g., repeated measures without accounting for it), the SE can be underestimated, producing intervals that are too narrow.
Is the interval answering the right question? A CI for a mean answers a different question than a CI for a proportion or for a difference between groups.

Now answer the exercise about the content:

Which statement best describes the correct long-run interpretation of a 95% confidence interval?

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You missed! Try again.

A confidence level describes the method’s long-run hit rate across repeated samples, not the probability that a single computed interval contains the fixed parameter.