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To determine an appropriate sample size n for estimating proportions, the equation below can be solved, where W represents the desired width of the confidence interval. The resulting sample size formula, is often applied with a conservative estimate of p (e.g., 0.5): = / for n, yielding the sample size Sample sizes for binomial proportions ...
In contrast, it is worth noting that other confidence interval may have coverage levels that are lower than the nominal , i.e., the normal approximation (or "standard") interval, Wilson interval, [8] Agresti–Coull interval, [13] etc., with a nominal coverage of 95% may in fact cover less than 95%, [4] even for large sample sizes.
The confidence interval can be expressed in terms of statistical significance, e.g.: "The 95% confidence interval represents values that are not statistically significantly different from the point estimate at the .05 level." [20] Interpretation of the 95% confidence interval in terms of statistical significance.
In the social sciences, a result may be considered statistically significant if its confidence level is of the order of a two-sigma effect (95%), while in particle physics and astrophysics, there is a convention of requiring statistical significance of a five-sigma effect (99.99994% confidence) to qualify as a discovery. [3]
For a confidence level, there is a corresponding confidence interval about the mean , that is, the interval [, +] within which values of should fall with probability . Precise values of z γ {\displaystyle z_{\gamma }} are given by the quantile function of the normal distribution (which the 68–95–99.7 rule approximates).
Comparison of the rule of three to the exact binomial one-sided confidence interval with no positive samples. In statistical analysis, the rule of three states that if a certain event did not occur in a sample with n subjects, the interval from 0 to 3/ n is a 95% confidence interval for the rate of occurrences in the population.
The confidence interval summarizes a range of likely values of the underlying population effect. Proponents of estimation see reporting a P value as an unhelpful distraction from the important business of reporting an effect size with its confidence intervals, [7] and believe that estimation should replace significance testing for data analysis ...
For example, f(x) might be the proportion of people of a particular age x who support a given candidate in an election. If x is measured at the precision of a single year, we can construct a separate 95% confidence interval for each age. Each of these confidence intervals covers the corresponding true value f(x) with confidence 0.