How to find confidence interval

The confidence interval in probability theory is used to estimate the mathematical expectation. In relation to the parameter analyzed by statistical methods, this is the intervalthat covers the value of this quantity with a given accuracy (the degree or level of reliability).

Let the random variable x is normally distributed and known standard deviation. Then the condence interval is equal to: m(x) – t·σ/√n < M(x) < m(x) + t·σ/√n, where m(x) – sample mean of sample x, t is the argument of the function Laplace, σ – standard deviation, n – sample size, M(x) is the mathematical expectation. The expression standing on the left and right of M(x) are called the confidence limits.

The Laplace function is used in the formula to determine the probability of a parameter value in this interval. Generally, when solving such problems requires either to calculate the function via argument, or Vice versa. The formula for finding features is a rather cumbersome integral, therefore for simplification of work with probabilistic models, use a ready-made table of values.

Example:Find confidence interval confidence level of 0.9 for the estimated sign of some General sample x, if it is known that the standard deviation σ is equal to 5, the sample mean m(x) = 20, size n = 100.

Solution:Identify the values involved in the formula are unknown to you. In this case, the mathematical expectation and the Laplace argument.

The statement says that the value of the function equal to 0.9, therefore, determine t from the table:Φ(t) = 0,9 → t = 1,65.

Substitute all known numbers into the formula and calculate the confidence limits:20 – 1,65·5/10 < M(x) < 20 + 1,65·5/1019,175 < M(x) < 20,825.