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Confidence Interval Formula:
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A confidence interval is a range of values that is likely to contain a population parameter with a certain level of confidence. It provides an estimate of the uncertainty associated with a sample statistic.
The calculator uses the confidence interval formula:
Where:
Explanation: The formula calculates the range within which the true population mean is likely to fall, with a specified level of confidence.
Details: Confidence intervals provide more information than point estimates alone by indicating the precision of an estimate and the uncertainty around it. They are widely used in statistical inference and hypothesis testing.
Tips: Enter the sample mean, z-score corresponding to your desired confidence level (common values: 1.96 for 95% CI, 2.576 for 99% CI), population standard deviation, and sample size. All values must be valid (sample size > 0).
Q1: What is the relationship between confidence level and z-score?
A: Higher confidence levels require larger z-scores. Common values: 90% CI = 1.645, 95% CI = 1.96, 99% CI = 2.576.
Q2: When should I use a t-score instead of a z-score?
A: Use a t-score when the population standard deviation is unknown and the sample size is small (typically n < 30). For larger samples, z-scores are appropriate.
Q3: How does sample size affect the confidence interval?
A: Larger sample sizes result in narrower confidence intervals, indicating more precise estimates of the population parameter.
Q4: Can I calculate confidence intervals for proportions?
A: Yes, but the formula differs: \( CI = p \pm z \times \sqrt{\frac{p(1-p)}{n}} \), where p is the sample proportion.
Q5: What does a 95% confidence interval actually mean?
A: It means that if we were to take many samples and build a confidence interval from each sample, approximately 95% of these intervals would contain the true population parameter.