Redshift Academy

Wolfram Alpha:         

  Search by keyword:  

Astronomy

-
-
-
-

Chemistry

-
-
-
-

Classical Mechanics

-

Classical Physics

-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-

Climate Change

-

Cosmology

-
-
-
-
-
-
-
-
-
-
-
-
-
-

Finance and Accounting

-
-
-
-
-
-
-
-
-

Game Theory

-

General Relativity

-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-

Group Theory

-
-
-
-
-
-

Lagrangian and Hamiltonian Mechanics

-
-
-
-
-
-

Macroeconomics

-
-
-

Mathematics

-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-

Mathjax

-

Microeconomics

-

Nuclear Physics

-
-

Particle Physics

-
-
-
-
-
-
-

Probability and Statistics

-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-

Programming and Computer Science

-
-
-
-
-
-

Quantitative Methods for Business

-

Quantum Computing

-
-
-

Quantum Field Theory

-
-
-
-
-

Quantum Mechanics

-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-

Semiconductor Reliability

-

Solid State Electronics

-
-
-
-
-

Special Relativity

-
-
-
-
-
-
-
-
-
-
-
-

Statistical Mechanics

-
-
-

String Theory

-
-
-
-
-
-

Superconductivity

-
-
-
-
-
-

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

-
-
-
-
-

The Standard Model

-
-
-
-
-
-
-
-
-
-

Topology

-

Units, Constants and Useful Formulas

-
Last modified: January 26, 2018

Sampling Distributions ---------------------- If we draw a sample size, n, from a given population and compute a statistic (mean, standard deviation, proportion) for each sample. The probability distribution of the statisitic is called a sampling distibution. Central Limit Theorem --------------------- For large enough sample sizes* the sample MEANS follow N(μ,σ/√n) where n is the sample size. In other words, the mean of all the sample means is equal to the population mean. This applies regardless of the distribution of the parent population. As the sample size is increased the standard deviation, skew and kurtosis of the sampling distribution decreases. The SD of the sampling distribution of the mean is called the STANDARD ERROR OF THE MEAN. SE = σ/√n = σsample means We can generalize this to: SE = (σ/√n)((N - n)/(N - 1)) Where N is the population size. For infinite N this reduces to: SE = σ/√n as before For small N it reduces to: SE = σ * A sample size ≥ 30 is generally considered to be the minimum for the CLT to apply. Distribution of Means --------------------- _ If x is the mean of a random sample of size n taken from a normal population having mean μ and variance σ2, then we can state the following: Large Sample Size (n > 30) -------------------------- _ z = (x - μ)/σ/√n Generally, the population standard deviation is not given. Under these circumstances it is necessary to compute it from the sample using:    _ s2 = Σi(x - xi)2/(n - 1) Why use n - 1? To answer that we need to look at biassed versus unbiassed estimators. Biassed versus Unbiassed Estimators ----------------------------------- Consider a small distribution: 5 7 12 N = 3 n = 2 μ = 8 σ2 = Σ(x - μ)2/N = 8.67 σ = 2.94 _   _ s2 = Σ(x - x)2/(n - 1) = Σ(x - x)2/1 _ Sample x s2 s s22 ------ --- ---- ---- ---- 5 5 5.0 0.0 0.00 0.00 5 7 6.0 2.0 1.41 0.23 5 12 8.5 24.5 4.95 2.83 7 5 6.0 2.0 1.41 0.23 7 7 7.0 0.0 0.00 0.00 7 12 9.5 12.5 3.54 1.44 12 5 8.5 24.5 4.95 2.83 12 7 9.5 12.5 3.54 1.44 12 12 12.0 0.0 0.00 0.00 Average 8.0 8.67 2.20 Notes: _ - x follows a t distribution (normal dostribution if the sample size is large). - s is a biassed estimator of σ - (n - 1)s22 follows the χ2 distribution. - A denominator of n gives the biassed estimator for σ2. - A denominator n - 1 gives the unbiassed estimator for σ2. Small Sample Size (n ≤ 30) -------------------------- If n ≤ 30 then we must use the t-statistic instead:   _ tv = (x - μ)/s/√n with v = n - 1 degrees of freedom. The t-distribution has the following properties: mean: μ = 0 variance: σ2 = v/(v - 2), where v is the degrees of freedom and v > 2 The variance is always greater than 1, although it is close to 1 when there are many degrees of freedom. With infinite degrees of freedom, the t distribution is the same as the standard normal distribution. Distribution of Proportions --------------------------- If p is the proportion (probability) of successes in the population, then: σp = √(p(1 - p)/n) The sampling distribution of p is a discrete rather than a continuous distribution. It is approximately normally distributed if n is fairly large and p is not close to 0 or 1. A general rule of thumb is that the approximation is good when: np and n(1 - p) are both ≥ 10 _ z = (p - μ)/σ/√n As before, generally, p and σ are not given for the original population. Under these circumstances it is necessary to compute them from the sample. Thus, p = psample Distribution of Variances ------------------------- If s2 is the variance of a random sample of size n taken from a normal poulation having the variance σ2, then χ2 = (n - 1)s22 with v = n - 1 degrees of freedom. The χ2 distribution has the following properties: mean: μ = v variance: σ2 = 2v Example: An optical firm purchases glass to be ground into lenses and past experience has shown that the variance of the refractive index = 1.26 x 10-4. A shipment is received and a sample of 20 pieces is pulled. The measured variance of the sample is 2.10 x 10-4. Should the sample be rejected? H0: σ2 = s2 H1: σ2 ≠ s2 χ2 = (20 - 1)2.10 x 10-4/1.26 x 10-4 = 31.66 From tables, χ20.05 for v = n - 1 = 19 is equal to 30.144. Since 31.66 > 30.144 the result is significant at the 0.05 level and there is sufficient reason to reject H0. F Distribution -------------- If s12 and s22 are the variances of independent random samples of size n1 and n2 taken from two normal populations having the same variance, then F = s12/s22 with v1 = n1 - 1 and v2 = n2 - 1 degrees of freedom. The F distribution has the following properties: mean: μ = v2/(v2 - 2) for v2 > 2. variance: σ2 = [2v22(v1 + v2 - 2)]/[v1(v2 - 2)2(v2 - 4)] for v2 > 4. Example: Suppose you randomly select 7 marbles from company 1's production line and 12 marbles from company 2's production line and measure their diameters. Assume you are given: s1 = 1.0 and s2 = 1.1 F = s12/s22 = 1/1.21 = 0.83 H0: σ1 = σ2 H1: σ1 ≠ σ2 From tables, F0.05 for v1 = n1 - 1 = 6 and v2 = n2 - 1 = 11 is equal to 3.0946. Since 0.83 < 3.0946 the result is not significant at the 0.05 and there is insufficient reason to reject H0.