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calculators.confidence-interval.criticalZValues
When to Use Each
출처 및 방법론
CI = x̄ ± z × (σ/√n)Confidence interval for mean
How to Use the Confidence Interval Calculator
The confidence interval calculator computes confidence intervals for means and proportions at various confidence levels. Visualize the interval with error bars and determine the sample size needed for your desired margin of error.
What is a Confidence Interval?
A confidence interval is a range of values that likely contains the true population parameter. A 95% confidence interval means: if we repeated the sampling process many times, about 95% of the intervals would contain the true value.
The Formula
Confidence Interval = Sample Statistic ± Margin of Error
Margin of Error = Critical Value × Standard Error
Components
- Sample Mean (x̄): The center of your interval
- Standard Deviation (s): Measure of spread
- Sample Size (n): Number of observations
- Confidence Level: Usually 90%, 95%, or 99%
Critical Values
Common z-values for different confidence levels:
- 90% confidence: z = 1.645
- 95% confidence: z = 1.96
- 99% confidence: z = 2.576
Use t-values instead of z-values for small samples (n < 30).
Interpreting Confidence Intervals
A 95% CI of (42.5, 47.5) means: "We are 95% confident that the true population mean is between 42.5 and 47.5."
Sample Size Determination
To achieve a specific margin of error:
n = (z × σ / E)²
where E is the desired margin of error.
Related Calculators
For p-values and hypothesis testing, use our p-value calculator. For standard deviation, try our standard deviation calculator.
자주 묻는 질문
95% confidence means: if we repeated the sampling and calculated intervals many times, about 95% of those intervals would contain the true population parameter. It does NOT mean theres a 95% chance the true value is in this specific interval - the true value either is or isnt in there.
A 95% CI of (42, 48) for mean height means we are 95% confident the population mean is between 42 and 48. Narrower intervals indicate more precision. If comparing groups, non-overlapping CIs suggest a significant difference.
Larger samples give narrower confidence intervals. Margin of error decreases with √n - quadrupling sample size halves the margin of error. This is why larger studies provide more precise estimates. Diminishing returns kick in at very large samples.
95% is standard in most fields. Use 99% when errors are costly (medical research, safety testing). Use 90% for preliminary analysis or when resources are limited. Higher confidence = wider interval. Choose before collecting data, not after.
Use n = (z × σ / E)², where z is the critical value, σ is standard deviation, and E is desired margin of error. For 95% CI with SD=10 and margin of error ±2: n = (1.96 × 10 / 2)² = 96. Always round up.
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