Estimation

Point estimates, confidence intervals, margin of error, and uncertainty.

Estimation uses sample data to approximate an unknown population quantity. A good estimate should include uncertainty, because a different random sample would usually give a different result.

Learning objectives

After this lesson you should be able to:

  • distinguish point estimates from interval estimates;
  • compute a margin of error from a critical value and standard error;
  • build introductory confidence intervals for means and proportions;
  • interpret confidence level correctly;
  • explain how sample size, spread, and confidence level affect interval width;
  • use edumath to verify confidence interval calculations.

Parameters and estimates

A parameter is an unknown number about a population, such as a true mean mu or true proportion p.

A point estimate is one sample-based guess, such as sample mean x-bar or sample proportion p-hat.

A point estimate is useful but incomplete because it does not show uncertainty.

Confidence intervals

A confidence interval has the form:

estimate ± margin of error

For many introductory normal-approximation settings:

margin of error = critical value * standard error

A 95% normal critical value is approximately 1.96.

Interpreting confidence level

A 95% confidence level means that if we repeatedly used the same method on many random samples, about 95% of the intervals would contain the true parameter.

It does not mean there is a 95% probability that one already-computed interval contains the fixed parameter. In introductory courses we often use informal language, but the more precise idea is long-run performance of the method.

What changes interval width?

Intervals become wider when:

  • confidence level increases;
  • data are more variable;
  • sample size is smaller.

Intervals become narrower when:

  • confidence level decreases;
  • data are less variable;
  • sample size increases.

Worked example 1: confidence interval for a mean

A sample of 36 observations has mean 50 and standard deviation 12. Use an approximate 95% interval for the population mean.

Step 1: Compute standard error.

SE = s / sqrt(n) = 12 / sqrt(36) = 12 / 6 = 2

Step 2: Compute margin of error.

ME = 1.96 * 2 = 3.92

Step 3: Build the interval.

50 ± 3.92 = (46.08, 53.92)

Step 4: Interpret.

We are 95% confident, using this method, that the population mean is between about 46.08 and 53.92, assuming the sampling conditions are appropriate.

Worked example 2: confidence interval for a proportion

In a sample of 100 people, 60 say yes. Estimate the population yes proportion with a 95% confidence interval.

Step 1: Find the sample proportion.

p-hat = 60 / 100 = 0.60

Step 2: Find standard error.

SE = sqrt(p-hat(1-p-hat)/n)
SE = sqrt(0.60 * 0.40 / 100)
SE = sqrt(0.0024)
SE is about 0.049

Step 3: Find margin of error.

ME = 1.96 * 0.049, about 0.096

Step 4: Build the interval.

0.60 ± 0.096 = (0.504, 0.696)

Interpretation: The data are consistent with a population yes proportion from about 50.4% to 69.6%, under the usual conditions.

Worked example 3: explain a wider interval

Two studies estimate the same mean with the same standard deviation. Study A has n=25; Study B has n=100.

Question: Which interval is wider?

Reasoning:

SE_A = s / sqrt(25) = s/5
SE_B = s / sqrt(100) = s/10

Study A has larger standard error, so its interval is wider.

Common pitfalls

  • Saying “there is a 95% probability the parameter is in this interval” without understanding the frequentist meaning.
  • Forgetting that a confidence interval depends on sampling conditions.
  • Reporting an interval without context or units.
  • Thinking a narrow interval is good if the sampling method was biased.
  • Confusing confidence level with the sample proportion.

Practice exercises

  1. What is a point estimate?
  2. A mean is 80, standard error is 4, and critical value is 1.96. Find the 95% interval.
  3. If confidence level increases from 90% to 99%, does the interval usually get wider or narrower?
  4. If sample size increases, what usually happens to margin of error?
  5. Interpret the interval (12.4, 18.8) for a population mean measured in hours.
  1. A single-number estimate of a population parameter, such as a sample mean.
  2. Margin of error =1.96(4)=7.84; interval (72.16, 87.84).
  3. Wider, because higher confidence requires a larger critical value.
  4. It usually decreases because standard error decreases.
  5. A careful interpretation: using the method and assumptions, we are confident that the population mean is between 12.4 and 18.8 hours.

Subtopic guided practice and checkpoints

Lab 1: estimate versus parameter

Guess first. Is p-hat=0.42 a parameter or statistic?

Guided exercise.

p-hat is computed from a sample, so it is a statistic and a point estimate of population proportion p.

Checkpoint. Is population mean mu a parameter or statistic?

Lab 2: build margin of error

Guess first. If standard error doubles, what happens to margin of error?

Guided exercise.

Since ME = critical value * SE, doubling SE doubles the margin of error.

Checkpoint. Find margin of error when critical value is 2 and SE=3.

Lab 3: interval interpretation

Guess first. Does a confidence interval prove the parameter is inside?

Guided exercise.

No. It gives a plausible range using a method with known long-run performance. The interval can miss the true parameter.

Checkpoint. What does 95% confidence describe: the single interval or the long-run method?

Lab 4: width decisions

Guess first. Which is wider: a 90% interval or a 99% interval from the same data?

Guided exercise.

The 99% interval is wider because it requires more confidence.

Checkpoint. What happens to interval width if data are more variable?

  1. Parameter.
  2. ME=6.
  3. The long-run method.
  4. It gets wider, all else equal.

Estimation guessing game

Using this lesson with edumath and SymPy

Use edumath to compute confidence intervals.

from edumath.statistics import confidence_interval_mean, confidence_interval_proportion

confidence_interval_mean([48, 52, 51, 49, 50], confidence_level=0.95)
ConfidenceInterval(estimate=50.0, lower=48.614096175650324, upper=51.385903824349676, margin_of_error=1.3859038243496775, confidence_level=0.95)
confidence_interval_proportion(60, 100)
ConfidenceInterval(estimate=0.6, lower=0.5039817664728938, upper=0.6960182335271061, margin_of_error=0.09601823352710616, confidence_level=0.95)

Use SymPy to rearrange the interval formula.

import sympy as sp

estimate, critical, se = sp.symbols("estimate critical se")
lower = estimate - critical * se
upper = estimate + critical * se
sp.simplify(upper - lower)

\(\displaystyle 2 \mathcal{criti} se\)