Math 2.0

Statistics Definition

Binomial proportion confidence interval

Builds on Confidence intervalBinomial distribution

Definition

Let n≥1n \ge 1, let XX have the binomial distribution with parameters nn and unknown p∈[0,1]p \in [0, 1], and let α∈(0,1)\alpha \in (0, 1). A binomial proportion confidence interval of level 1−α1 - \alpha is an interval [L(X),U(X)][L(X), U(X)] with endpoints computed from XX alone. Its coverage function is C(p)=Pp(L(X)≤p≤U(X))C(p) = P_p\big(L(X) \le p \le U(X)\big). It is a confidence interval of level at least 1−α1 - \alpha when C(p)≥1−αC(p) \ge 1 - \alpha for every p∈[0,1]p \in [0, 1]. It is an approximate interval when C(p)C(p) tends to 1−α1 - \alpha as n→∞n \to \infty, for each fixed p∈(0,1)p \in (0, 1).

Remarks

Let p^=X/n\hat p = X/n and let zz be the 1−α/21 - \alpha/2 quantile of the standard normal law. The Wald interval p^±zp^(1−p^)/n\hat p \pm z \sqrt{\hat p(1 - \hat p)/n} is an approximate interval, by the normal approximation to the binomial law. The Wilson interval is the set of pp with ∣p^−p∣≤zp(1−p)/n\abs{\hat p - p} \le z \sqrt{p(1 - p)/n}; it is also approximate, with better coverage for small nn. The Clopper–Pearson interval has level at least 1−α1 - \alpha for every pp; its endpoints are quantiles of beta laws. Non-example: the Wald interval is not of level at least 1−α1 - \alpha for all pp. If X=0X = 0, it is the single point {0}\set{0}, so for small p>0p > 0 the coverage C(p)C(p) is close to Pp(X≥1)=1−(1−p)nP_p(X \ge 1) = 1 - (1 - p)^n, which tends to 00 as p→0p \to 0.

Reading path

To understand this page from the ground up, read these 78 pages in order. The foundations come first.

  1. Binomial coefficient definition
  2. Construction of the reals from Cauchy sequences concept
  3. Element definition
  4. Integers object
  5. Ordered field definition
  6. Proposition concept
  7. Random experiment concept
  8. Rational numbers object
  9. Real numbers object
  10. Set definition
  11. Ring of integers statement
  12. Disjunction definition
  13. Implication definition
  14. Predicate definition
  15. Bernoulli trial concept
  16. Ordered pair definition
  17. Union definition
  18. Sample space definition
  19. Existential quantifier definition
  20. The rationals form a field statement
  21. Universal quantifier definition
  22. Cartesian product definition
  23. Subset definition
  24. The rationals form an ordered field statement
  25. Uniqueness quantifier definition
  26. Binary relation definition
  27. Universal set definition
  28. Power set definition
  29. Order on the reals definition
  30. Complement definition
  31. Function definition
  32. Interval definition
  33. Sigma-algebra definition
  34. Indicator function definition
  35. Injective function definition
  36. Preimage of a set definition
  37. Event definition
  38. Countable set definition
  39. Borel sigma-algebra definition
  40. Probability axioms concept
  41. Probability measure definition
  42. Dirac measure definition
  43. Probability space definition
  44. Probability distribution definition
  45. Conditional probability definition
  46. Random variable definition
  47. Independence of two events definition
  48. Discrete random variable definition
  49. Standard score definition
  50. Cumulative distribution function definition
  51. Law of a random variable definition
  52. Random variable with a density definition
  53. Mutual independence of events definition
  54. Independence of two random variables definition
  55. Probability mass function definition
  56. Joint probability distribution definition
  57. Expected value definition
  58. Identically distributed random variables definition
  59. Probability density function definition
  60. Independent sigma-algebras definition
  61. Binomial distribution definition
  62. Variance definition
  63. Marginal distribution definition
  64. Linearity of expectation statement
  65. Density of the normal distribution definition
  66. Independence of random variables definition
  67. Independence means the joint law is the product of the marginals statement
  68. Covariance definition
  69. Normal distribution definition
  70. Expectation of a product of independent random variables statement
  71. Independent and identically distributed random variables definition
  72. Variance of a sum statement
  73. Independent random variables are uncorrelated statement
  74. Statistic definition
  75. Variance of a sum of independent random variables statement
  76. Sample mean definition
  77. Confidence interval definition
  78. Binomial proportion confidence interval definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval
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