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Applications of Confidence Intervals

Key Terms

A CI gives a range of plausible values for the true proportion p
If a claimed value lies outside the CI, it is inconsistent with the data
If a claimed value lies inside the CI, the data are consistent with the claim
If two CIs do not overlap, there is evidence of a genuine difference between groups
If two CIs overlap, we cannot conclude a difference exists
Always interpret results in context — state what the interval means for the real-world situation
Decision Framework:
Claimed value p0 outside CI → data are inconsistent with the claim
Claimed value p0 inside CI → data are consistent with the claim

Comparing Two Groups:
CI1 and CI2 do not overlap → evidence of real difference
CI1 and CI2 overlap → insufficient evidence to conclude a difference
Worked Example: A cereal company claims 70% of children prefer its brand. In a random survey of 300 children, 195 prefer it. Does the data support the claim at the 95% level?

p̂ = 195/300 = 0.65
SE = √(0.65 × 0.35/300) = √(0.000758) ≈ 0.02754
ME = 1.96 × 0.02754 ≈ 0.05398
95% CI: (0.596, 0.704)

The claimed value of 0.70 lies inside the interval (0.596, 0.704), so the data are consistent with the company’s claim. We cannot conclude the claim is wrong.
Hot Tip: Saying the interval “supports” a claim is different from proving it true. A CI only tells us whether the claim is plausible — many other values are also plausible within the interval.

Using CIs to Assess Claims

One of the most common applications of confidence intervals is to evaluate whether a claimed population proportion is supported by sample data. The logic is:

  • Build a CI from your sample data
  • Check whether the claimed value p0 falls inside the interval
  • If yes: the data are consistent with the claim (the claim is plausible)
  • If no: the data contradict the claim (the claim is implausible at this confidence level)

This is closely related to hypothesis testing — a claim that falls outside a 95% CI would be rejected in a two-tailed hypothesis test at the 5% significance level.

Comparing Two Groups

When comparing proportions from two different groups, construct a CI for each group separately and compare:

  • Non-overlapping CIs: Strong evidence that the true proportions differ
  • Overlapping CIs: Insufficient evidence to conclude a difference (though this is an informal method — a formal two-sample test is more rigorous)

Interpreting in Context

Every confidence interval should be interpreted in the context of the problem. A good interpretation:

  1. States the confidence level
  2. Names the quantity being estimated (the true population proportion of...)
  3. Gives the interval bounds
  4. Draws a conclusion relevant to the question asked
Strategy: Don’t just state the interval — say what it means. “We are 95% confident the true proportion of voters who support the policy is between 48% and 56%” is a complete interpretation.

Mastery Practice

See Answers ➔
  1. Fluency A 95% CI for the proportion of students who pass a test is (0.72, 0.84). A teacher claims the pass rate is 90%. Is this claim consistent with the data? Explain.
  2. Fluency A survey of 250 adults finds that 140 exercise regularly. Construct a 95% CI and state whether a claim of p = 0.50 is supported.
  3. Fluency In a quality control check of 400 units, 32 are faulty. Construct a 99% CI for the true defect proportion. The company claims its defect rate is under 10% — is this consistent with the data?
  4. Fluency Write a complete contextual interpretation of the CI (0.61, 0.73) constructed from a survey of 500 people asked whether they support a new road.
  5. Understanding A government department conducts surveys in two cities. In City X (n = 400), 236 residents support a new park. In City Y (n = 400), 212 support it. Construct 95% CIs for each city and determine whether there is evidence of a difference in support levels.
  6. Understanding A researcher claims that a new teaching method produces a pass rate of at least 80%. A trial with 120 students has 88 passing. Construct a 95% CI. Does the interval support the researcher’s claim?
  7. Understanding A news article states: “A poll of 1000 voters gives candidate A 53% support, with a margin of error of ±3.1%.” Verify the margin of error and explain whether candidate A can claim majority support.
  8. Understanding Two surveys about internet usage are conducted. Survey 1: n = 200, p̂ = 0.74. Survey 2: n = 800, p̂ = 0.68. Construct 95% CIs for both. Do the intervals overlap? What does this suggest?
  9. Problem Solving A pharmaceutical company claims its new drug relieves symptoms in more than 60% of patients. A clinical trial of 180 patients finds 114 experience relief.
    • (a) Construct a 95% CI for the true proportion.
    • (b) Does the data support the company’s claim? Justify.
    • (c) Regulators require strong evidence (99% CI). Does the claim still hold?
  10. Problem Solving A sports analyst compares the proportion of home wins in two football leagues. League A: 145 home wins out of 240 games. League B: 198 home wins out of 340 games.
    • (a) Construct 95% CIs for the home win proportion in each league.
    • (b) Do the intervals overlap? What conclusion can you draw about whether the leagues have different home-ground advantages?
    • (c) Both leagues have a long-standing belief that home advantage produces wins “more than half the time.” Does the data from each league support this belief?

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