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Confidence intervalsStatistics and probabilityCollegeabout 1:18

A sample of 36 has mean 100; the population standard deviation is 15. Find the 95% confidence interval for the mean: animated lesson

A narrated, step-by-step confidence intervals video lesson with captions. Final answer: 95% CI: (95.1001, 104.9); margin of error 4.89991.

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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 (0.5 MB) · Captions (WebVTT)

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1. The problem
Worked example
A sample of 36 has mean 100; the population standard deviation is 15. Find the 95% confidence interval for the mean.
2. Step by step
1. $SE = \frac{\sigma}{\sqrt n} = \frac{15}{\sqrt{36}} \approx 2.5$
  Standard error of the mean
2. $z^* = 1.95996$
  Critical value for 95% confidence
3. $E = z^* \times SE = 1.95996 \times 2.5 \approx 4.89991$
  Margin of error
4. $100 \pm 4.89991 = (95.1001,\ 104.9)$
  Interval: estimate ± margin
3. The answer
$(95.1001,\ 104.9)$
100 ± 4.89991
4. Check
- The interval is centred on the estimate: (95.1001 + 104.9) / 2 = 100.

Transcript

The problem. Here is the problem: A sample of 36 has mean 100; the population standard deviation is 15. Find the 95 percent confidence interval for the mean. Build a 95 percent confidence interval for the population mean μ.

Step by step. Standard error of the mean. Critical value for 95 percent confidence. The middle 95 percent leaves 2.5 percent in each tail. σ is treated as known, so the standard normal distribution gives the critical value. Margin of error. Interval: estimate plus or minus margin.

The answer. So the answer is 95 percent CI: (95.1001, 104.9); margin of error 4.89991. We are 95 percent confident that the population mean μ lies between 95.1001 and 104.9. The confidence describes the method: about 95 percent of intervals built this way from repeated samples would capture the true value.

Check. The interval is centred on the estimate: (95.1001 plus 104.9) / 2 equals 100.

About confidence intervals

Build confidence intervals for a mean or a proportion.

A sample of 36 has mean 100; the population standard…