Confidence Interval Calculator
Calculate the confidence interval for a population mean using the sample mean, standard deviation, sample size, and z-value.
Lower Bound
46.7333
Lower Bound vs Sample Mean
Formula
## How to Calculate a Confidence Interval ### Formula **CI = x_bar +/- z * (sigma / sqrt(n))** The confidence interval gives a range of plausible values for the population mean. The standard error (sigma/sqrt(n)) measures how much the sample mean varies across samples. The z-value determines the confidence level: 1.645 for 90%, 1.96 for 95%, 2.576 for 99%.
Exemplo Resolvido
Sample mean = 50, SD = 10, n = 36. Build a 95% confidence interval.
- 01Standard Error = 10 / sqrt(36) = 10 / 6 = 1.6667
- 02Margin of Error = 1.96 * 1.6667 = 3.2667
- 03Lower bound = 50 - 3.2667 = 46.7333
- 04Upper bound = 50 + 3.2667 = 53.2667
- 0595% CI: (46.7333, 53.2667)
Perguntas Frequentes
What does 95% confidence mean?
If you repeated the sampling and CI construction many times, approximately 95% of those intervals would contain the true population mean. It does not mean there is a 95% probability the true mean is in this specific interval.
When should I use z vs. t?
Use z when the population standard deviation is known or the sample size is large (n >= 30). Use the t-distribution when the population SD is unknown and the sample is small.
How does sample size affect the interval?
Larger sample sizes produce narrower confidence intervals because the standard error decreases as sqrt(n) increases. Quadrupling the sample size halves the margin of error.
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