Confidence Interval for the Mean
xฬ ยฑ z*(ฯ/โn) or xฬ ยฑ t*(s/โn)
A confidence interval provides a range of plausible values for the population mean based on sample data. When the population standard deviation is known, use the z-interval; when it is unknown and estimated by s, use the t-interval. The confidence level (e.g., 95%) reflects how often the procedure produces intervals that contain the true mean.
Variables
The point estimate of the population mean
The z* or t* value corresponding to the desired confidence level
Population standard deviation (ฯ) or sample standard deviation (s)
The number of observations in the sample
Example Calculation
Scenario
A sample of 36 light bulbs has a mean lifetime of 1200 hours with a known population standard deviation of 120 hours. Construct a 95% confidence interval.
Given Data
Calculation
xฬ ยฑ z*(ฯ/โn) = 1200 ยฑ 1.96(120/โ36) = 1200 ยฑ 1.96(20) = 1200 ยฑ 39.2
Result
(1160.8, 1239.2) hours
Interpretation
We are 95% confident that the true population mean lifetime of the light bulbs falls between 1160.8 and 1239.2 hours. If we repeated this procedure many times, about 95% of the resulting intervals would contain the true mean.
When to Use This Formula
- โEstimating the population mean with a quantified level of uncertainty
- โReporting results in research papers alongside point estimates
- โDetermining whether a hypothesized value is plausible for the population mean
- โPlanning sample sizes by inverting the margin of error formula
Common Mistakes
- โUsing z* when the population standard deviation is unknown (should use t*)
- โInterpreting the interval as a probability statement about the specific interval rather than the procedure
- โForgetting that larger confidence levels produce wider intervals
- โNot checking that the sample is random and the sampling distribution is approximately normal
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Common questions about this formula
It means that if you repeated the sampling and interval construction process many times, approximately 95% of those intervals would contain the true population mean. It does not mean there is a 95% probability that this particular interval contains the mean; the true mean is fixed, not random.
Use z* when the population standard deviation (ฯ) is known and the population is normal or n is large. Use t* when ฯ is unknown and you estimate it with the sample standard deviation (s). In practice, t-intervals are far more common because ฯ is rarely known.