Interpreting confidence intervals and linking them to tests

What 95% means

μ is a fixed (unknown) number; the interval is random because it depends on the sample. If many samples were taken and an interval calculated from each, about 95% of the intervals would contain μ.

Once calculated, a particular interval either contains μ or it does not.

Several intervals

Intervals from independent samples are independent events, each containing μ with probability equal to the confidence level. So the number of 95% intervals out of m that contain μ is B(m, 0.95).

P(two 95% intervals both contain μ) = 0.95² = 0.9025. P(exactly one of the two does) = 2 × 0.95 × 0.05 = 0.095.

Link with hypothesis tests

A two-tailed test of H0: μ = μ0 at the 5% level rejects H0 exactly when μ0 lies outside the 95% confidence interval (similarly 1% and the 99% interval).

Example: the 95% interval is (29.63, 32.77). H0: μ = 33 is rejected at the 5% level; H0: μ = 32 is not.

Worked example

Ten independent 90% confidence intervals for μ are calculated. Find the probability that at least 9 of them contain μ.

1. Number containing μ, N ~ B(10, 0.9).
2. P(N ≥ 9) = 0.9¹⁰ + 10 × 0.9⁹ × 0.1

Answer: P = 0.7361

Check yourself

1. Two independent 95% confidence intervals for μ are calculated. Find the probability that both contain μ. Give your answer to 4 decimal places.
   Show answer

   0.9025. 0.95 × 0.95 = 0.9025.

2. Two independent 95% confidence intervals for μ are calculated. Find the probability that exactly one of them contains μ.
   Show answer

   0.095. 2 × 0.95 × 0.05 = 0.095.

3. 40 independent 95% confidence intervals for μ are calculated. How many would you expect to contain μ?
   Show answer

   38. 40 × 0.95 = 38.

4. Ten independent 90% confidence intervals for μ are calculated. Find the probability that at least 9 of them contain μ. Give your answer to 4 decimal places.
   Show answer

   0.7361. 0.9¹⁰ + 10(0.9⁹)(0.1) = 0.7361.

5. A 95% confidence interval for μ is (29.6, 32.8). Which interpretation is correct?
   • If many such intervals were calculated, about 95% of them would contain μ
   • There is a 95% probability that μ lies between 29.6 and 32.8
   • 95% of the data lie between 29.6 and 32.8
   • μ changes, and lies in the interval 95% of the time
   Show answer

   If many such intervals were calculated, about 95% of them would contain μ. The 95% refers to the method: the proportion of intervals that capture the fixed μ.

6. A 95% confidence interval for μ is (29.6, 32.8). A two-tailed test at the 5% level is carried out of H0: μ = 33. What is the outcome?
   • Do not reject H0, because 33 is close to the interval
   • Reject H0, because 33 lies inside the interval
   • Reject H0, because 33 lies outside the interval
   • The interval cannot be used for a test
   Show answer

   Reject H0, because 33 lies outside the interval. Values outside the 95% interval are exactly those rejected by a two-tailed 5% test.

7. A 99% confidence interval for μ is (14.2, 17.9). What is the outcome of a two-tailed 1% test of H0: μ = 15?
   • Do not reject H0, because 15 lies inside the interval
   • Reject H0, because 15 lies inside the interval
   • Reject H0, because 15 is not the centre
   • The test needs a 95% interval
   Show answer

   Do not reject H0, because 15 lies inside the interval. 15 is a plausible value of μ at this level, so there is insufficient evidence to reject H0.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S3

Last updated 8 November 2026. Spotted a mistake? Tell us.