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95 CI Confidence Intervals.md

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Confidence Intervals

Wrong Interpretation

When reporting a CI:

  • WRONG: “There is a 95% probability that the mean is between 25.1 and 32.6”. Either μ is in that interval or not; there is no probability associated with it.
  • GOOD: “I am 95% confident that the mean is between 25.1 and 32.6”.

First, let’s assume we have access to the population. (This is, of course, never the case. Otherwise, we wouldn’t have to estimate a parameter but could compute it precisely.)

Then, if we draw a very large number of samples from the distribution, and apply our confidence interval method to these samples, 95% of the confidence intervals would contain the actual value.

A confidence interval is an interval associated with a parameter and is a frequentist concept. The parameter is assumed to be non-random but unknown, and the confidence interval is computed from data.

Because the data are random, the interval is random. A 95% confidence interval will contain the true parameter with probability 0.95. That is, with a large number of repeated samples, 95% of the intervals would contain the true parameter.

95% confidence means that if you were to repeat the sampling process 100 times, 95 of those samples would produce a confidence interval that contains the true parameter.

How to compute

CI = sample mean ± (critical value) * (standard error)

$$ CI = \bar{x} \pm z_{\alpha/2} \cdot \frac{s}{\sqrt{n}} $$

  • $\bar{x}$ is the sample mean
  • $z_{\alpha/2}$ is the critical value from the standard normal distribution
  • $s$ is the sample standard deviation
  • $n$ is the sample size

You can get the critical value from the standard normal distribution from a z-table or t-table.

Choose between z-table and t-table based on these criteria:

  • Use z-table when:
    • Sample size is large (n ≥ 30), OR
    • Population standard deviation (σ) is known
  • Use t-table when:
    • Sample size is small (n < 30), AND
    • Population standard deviation is unknown (using sample standard deviation s instead)

Statistical Significance

The difference of two measurements is statistically significant if confidence intervals do not overlap.

However we cannot say that results are not statistically significant if confidence intervals overlap.