07+Sampling+distribution

The probability that the auditors will not certify

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Unformatted text preview: P1 and P2 are shown as the square of the area under the distribution function less than z = −2 and greater than z = 2 respectively. The probability that the auditors will not certify the report is P = P1 + P2 = 0.0456. T P1 P2 E −2 0 2 z Pic. 1 4 A. Zhensykbaev Sampling distributions Other sampling distributions. Student’s t-distribution is defined by the function T= X −µ √, S/ n where S 2 is the sample variance 1 S= n−1 n 2 (xi − X )2 . i=1 If x is normally distributed the random variable will possess t-distribution. T ' Normal © t-distribution E 0 z, t Pic. 2 Fisher’s F -distribution. Suppose, we have two independent samples X = {x1 , ..., xn } and Y = {y1 , ..., ym }. If the random variables X and Y are normally distributed the the random variable F F= 1 n−1 1 m−1 n (xi − X )2 i=1 m (yi − Y )2 i=1 will possess Fisher’s F -distribution with n − 1 and m − 1 degrees of freedom (n − 1 is the numerator degrees of freedom, m − 1 is the denominator degrees of freedom ). 5 A. Zhensykbaev Sampling distributions T E Pic. 3 Estimation with confidence interval. We consider the methods of estimating of a population mean using large and small random samples. Large-sample estimation of a population mean. An interval estimator or confidence interval is a formula of calculation an interval estimate based on sample data. The probability that a randomly selected confidence interval encloses the population parameter is called the confidence coefficient. The confidence coefficient expressed as a percentage is called the confidence level. Let us consider normal random variable. Recall that due to the Central Limit Theorem for large samples the probability distribution of x is approxiσ mately normal: µ = x, with mean µ, and standard deviation σx = √n , ˆ 2σ µ ≈ x ± 2σx = x ± √ , n where µ - the population mean, σ - the population standard deviation and n is the sample size. Fix some number α ∈ [0, 1]. The confidence interval with the confidenc...
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