Definition D1
When two values L(X) and U(X), computed from the sample X = (X₁, …, Xₙ) alone, satisfy L(X) ≤ U(X), the interval I(X) = [L(X), U(X)] is called an interval estimate.
Proof.
(1)μ is a value belonging to the rule that generates the data, and it is not observed. μ is not used in computing L and U.convention
(2)L and U are functions of X, so when X is a random variable, L(X) and U(X) are random variables as well.(1), A4
(3)Therefore I(X) is not a single interval but an interval that varies with the sample.(2), A4
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Definition D2
For the event C = 'L(X) ≤ μ ≤ U(X)', when P(C) = 0.95, the interval I(X) is called a confidence interval at confidence level 95%.
Proof.
(1)Since 0 ≤ 0.95 ≤ 1, the value 0.95 is admissible as a probability.A1, N
(2)C is an event that occurs for some samples and not for others, and P attaches to the procedure of drawing the sample.D1, A4
(3)This definition speaks only of the procedure as a whole; it says nothing about any one particular sample.(2), D1
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Theorem T1
Once a sample is obtained and X = x, the interval becomes two fixed numbers l, u. Then 'the probability that μ lies in that interval' is 0 or 1, not 0.95.
Proof.
(1)Suppose X = x. A function's value is uniquely determined once the input is fixed, so L(x) = l and U(x) = u are fixed numbers.assumption, D1, A5
(2)μ is a fixed constant.A6
(3)Therefore 'l ≤ μ ≤ u' is a relation among three fixed numbers, and by the excluded middle it is true or false.(1), (2), A7
(4)If true, it has certainly occurred, so the probability is 1; if false, it cannot occur, so the probability is 0.(3), A2
(5)Since 0 ≠ 0.95 and 1 ≠ 0.95, that probability is not 0.95.(4), N
(6)l and u are observed but μ is not, so whether it is 0 or 1 cannot be known.(1), (4), D1
(7)Not knowing which and being 0.95 are different statements.(5), (6)
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Theorem T2
Among intervals obtained by repeating the same procedure, the proportion covering μ approaches 0.95.
Proof.
(1)Each repetition yields a new sample X and a new interval I(X).D1
(2)In each repetition, C either occurs or does not, and P(C) = 0.95.D2, A3
(3)By the law of large numbers, as the repetitions increase, the proportion in which C occurred approaches P(C) = 0.95.(2), A8
(4)Therefore 0.95 is a number attached to the procedure, not to any single interval.(3), D1, D2
(5)Repeating 100 times and obtaining 100 intervals, about 95 of them cover μ. Which 95 they are is not determined.(3), N
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Corollary C
'The probability that μ lies in this interval is 0.95' refers to something different from the probability D2 speaks of.
Proof.
(1)The probability of D2 attaches to the event C, and C is an event about the sample X.D2, D1
(2)For the interval [l, u] obtained after sampling, 'the probability that μ lies inside it' is 0 or 1.T1
(3)Since 0.95 is neither 0 nor 1, the word 'probability' in the two sentences does not refer to the same number.(1), (2), N
(4)Therefore 95% is not the degree to which one obtained interval contains μ, but the proportion of intervals covering μ when the procedure is repeated.(3), T2
(5)'Under repetition, 95% cover' follows from D2; 'the probability that this interval covers is 95%' does not.(4), T2, D2
(6)What the confidence interval says is a property of the procedure; what it does not say is a property of the one interval obtained.(4), (5)
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