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Confidence intervalsStatistics and probabilityCollegeabout 1:38

A sample of 400 has a sample proportion of 0.6. Find the 90% confidence interval for the proportion: animated lesson

A narrated, step-by-step confidence intervals video lesson with captions. Final answer: 90% CI: (0.559709, 0.640291); margin of error 0.0402905.

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Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.7 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Worked example
A sample of 400 has a sample proportion of 0.6. Find the 90% confidence interval for the proportion.
2. Steps, part 1
1. $\hat p = 0.6$
  Sample proportion
2. $SE = \sqrt{\frac{\hat p(1-\hat p)}{n}} = \sqrt{\frac{0.6(1-0.6)}{400}} \approx 0.0244949$
  Standard error
3. $z^* = 1.64485$
  Critical value for 90% confidence
4. $E = z^* \times SE = 1.64485 \times 0.0244949 \approx 0.0402905$
  Margin of error
3. Steps, part 2
1. $0.6 \pm 0.0402905 = (0.559709,\ 0.640291)$
  Interval: estimate ± margin
4. The answer
$(0.559709,\ 0.640291)$
0.6 ± 0.0402905
5. Check
- The interval is centred on the estimate: (0.559709 + 0.640291) / 2 = 0.6.
- Large-sample condition met: np̂ ≈ 240 and n(1 − p̂) ≈ 160 are both at least 10.

Transcript

The problem. Here is the problem: A sample of 400 has a sample proportion of 0.6. Find the 90 percent confidence interval for the proportion. Build a 90 percent confidence interval for the population proportion p.

Steps, part 1. Sample proportion. Standard error. Critical value for 90 percent confidence. The middle 90 percent leaves 5 percent in each tail. Large-sample condition met: np̂ approximately 240 and n(1 minus p̂) approximately 160 are both at least 10. Margin of error.

Steps, part 2. Interval: estimate plus or minus margin.

The answer. So the answer is 90 percent CI: (0.559709, 0.640291); margin of error 0.0402905. We are 90 percent confident that the population proportion p lies between 0.559709 and 0.640291. The confidence describes the method: about 90 percent of intervals built this way from repeated samples would capture the true value.

Check. The interval is centred on the estimate: (0.559709 plus 0.640291) / 2 equals 0.6. Large-sample condition met: np̂ approximately 240 and n(1 minus p̂) approximately 160 are both at least 10.

About confidence intervals

Build confidence intervals for a mean or a proportion.

A sample of 400 has a sample proportion of 0.6. Find the…