Calculate confidence intervals

Calculate confidence intervals

Understanding Confidence Intervals

  • A confidence interval is a range of values, derived from a data sample, which is likely to contain the value of an unknown population parameter.
  • Confidence intervals offer an interval estimate of population parameters rather than a single-point estimate, providing a range that we estimate contains the true population value most of the time.
  • The confidence level, represented as a percentage (often 95% or 99%), represents the degree of certainty that the confidence interval contains the true population parameter.

Construction of Confidence Intervals

  • The formula to calculate a confidence interval is: sample statistic ± margin of error.
  • The margin of error is calculated as the standard error multiplied by the critical value (also known as the z-value) for the desired confidence level.
  • The standard error is a measure of the variability or dispersion of the sampling distribution. It depends on both the standard deviation of the sample and the sample size.
  • The critical value can be found using a standard normal distribution (z-distribution) table, and it corresponds to the desired confidence level.

Interpretation of Confidence Intervals

  • The interpretation of a confidence interval is probabilistic, meaning it reflects the probability that the interval will contain the true population parameter after repeated sampling.
  • A 95% confidence interval does not mean that there is a 95% probability that the specific interval calculated from a given sample contains the true population parameter. Instead, it means if we were to sample repeatedly from a population, 95% of the intervals estimated would contain the population parameter.

Applications and Limitations of Confidence Intervals

  • Confidence intervals are widely used in scientific research, quality control, and survey sampling to provide a range of plausible values for an unknown parameter.
  • However, they are only as accurate as the assumptions that go into them. Faulty assumptions about the population distribution, sampling process, or independence of observations can lead to misleading confidence intervals.
  • Confidence intervals cannot account for sampling bias, and while increasing the sample size can increase the precision of the interval, it will not correct for bias.