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Confidence intervalsStatistics and probabilityCollegeabout 1:16

A sample of 64 has mean 50; the population standard deviation is 8. Find the 90% confidence interval for the mean: animated lesson

A narrated, step-by-step confidence intervals video lesson with captions. Final answer: 90% CI: (48.3551, 51.6449); margin of error 1.64485.

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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 64 has mean 50; the population standard deviation is 8. Find the 90% confidence interval for the mean.
2. Step by step
1. $SE = \frac{\sigma}{\sqrt n} = \frac{8}{\sqrt{64}} \approx 1$
  Standard error of the mean
2. $z^* = 1.64485$
  Critical value for 90% confidence
3. $E = z^* \times SE = 1.64485 \times 1 \approx 1.64485$
  Margin of error
4. $50 \pm 1.64485 = (48.3551,\ 51.6449)$
  Interval: estimate ± margin
3. The answer
$(48.3551,\ 51.6449)$
50 ± 1.64485
4. Check
- The interval is centred on the estimate: (48.3551 + 51.6449) / 2 = 50.

Transcript

The problem. Here is the problem: A sample of 64 has mean 50; the population standard deviation is 8. Find the 90 percent confidence interval for the mean. Build a 90 percent confidence interval for the population mean μ.

Step by step. Standard error of the mean. Critical value for 90 percent confidence. The middle 90 percent leaves 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 90 percent CI: (48.3551, 51.6449); margin of error 1.64485. We are 90 percent confident that the population mean μ lies between 48.3551 and 51.6449. 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: (48.3551 plus 51.6449) / 2 equals 50.

About confidence intervals

Build confidence intervals for a mean or a proportion.

A sample of 64 has mean 50; the population standard…