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Simple Random Sampling

Simple random sampling refers to any sampling method that has the following properties.

• The population consists of N objects.
• The sample consists of n objects.
• If all possible samples of n objects are equally likely to occur, the sampling method is called simple random sampling.

An important benefit of simple random sampling is that it allows researchers to use statistical methods to analyze sample results. For example, given a simple random sample, researchers can use statistical methods to define a confidence interval around a sample mean. Statistical analysis is not appropriate when non-random sampling methods are used.

There are many ways to obtain a simple random sample. One way would be the lottery method. Each of the N population members is assigned a unique number. The numbers are placed in a bowl and thoroughly mixed. Then, a blind-folded researcher selects n numbers. Population members having the selected numbers are included in the sample.

Important Statistics Formulas

Statistics Tutorial: Important Statistics Formulas

This web page presents statistics formulas described in the Stat Trek tutorials. Each formula links to a web page that explains how to use the formula.

Parameters

• Population mean = μ = ( Σ X_i ) / N
• Population standard deviation = σ = sqrt [ Σ ( X_i - μ )² / N ]
• Population variance = σ² = Σ ( X_i - μ )² / N
• Variance of population proportion = σ_P² = PQ / n
• Standardized score = Z = (X - μ) / σ
• Population correlation coefficient =
  ρ = [ 1 / N ] * Σ { [ (X_i - μ_X) / σ_x ] * [ (Y_i - μ_Y) / σ_y ] }

Statistics

Unless otherwise noted, these formulas assume simple random sampling.

• Sample mean = x = ( Σ x_i ) / n
• Sample standard deviation = s = sqrt [ Σ ( x_i - x )² / ( n - 1 ) ]
• Sample variance = s² = Σ ( x_i - x )² / ( n - 1 )
• Variance of sample proportion = s_p² = pq / (n - 1)
• Pooled sample proportion = p = (p₁ * n₁ + p₂ * n₂) / (n₁ + n₂)
• Pooled sample standard deviation =
  
  
  • s_p = sqrt [ (n₁ - 1) * s₁² + (n₂ - 1) * s₂² ] / (n₁ + n₂ - 2) ]
• Sample correlation coefficient = 
  
  
  • r = [ 1 / (n - 1) ] * Σ { [ (x_i - x) / s_x ] * [ (y_i - y) / s_y ] }

Correlation

• Pearson product-moment correlation = r = Σ (xy) / sqrt [ ( Σ x² ) * ( Σ y² ) ]
• Linear correlation (sample data) =
  r = [ 1 / (n - 1) ] * Σ { [ (x_i - x) / s_x ] * [ (y_i - y) / s_y ] }
• Linear correlation (population data) =
  
  
  • ρ = [ 1 / N ] * Σ { [ (X_i - μ_X) / σ_x ] * [ (Y_i - μ_Y) / σ_y ] }

Simple Linear Regression

• Simple linear regression line: y = b₀ + b₁x
• Regression coefficient = b₁ = Σ [ (x_i - x) (y_i - y) ] / Σ [ (x_i - x)²]
• Regression slope intercept = b₀ = y - b₁ * x
• Regression coefficient = b₁ = r * (s_y / s_x)
• Standard error of regression slope =
  s_b1 = sqrt [ Σ(y_i - y_i)² / (n - 2) ] / sqrt [ Σ(x_i - x)² ]

Counting

• n factorial: n! = n * (n-1) * (n - 2) * . . . * 3 * 2 * 1. By convention, 0! = 1.
• Permutations of n things, taken r at a time: _nC_r = n! / (n - r)!
• Combinations of n things, taken r at a time: _nC_r = n! / r!(n - r)! = _nP_r / r!

Probability

• Rule of addition: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
• Rule of multiplication: P(A ∩ B) = P(A) P(B|A)
• Rule of subtraction: P(A') = 1 - P(A)

Random Variables

In the following formulas, X and Y are random variables, and a and b are constants.

• Expected value of X = E(X) = μ_x = Σ [ x_i * P(x_i) ]
• Variance of X =
  Var(X) = σ² = Σ [ x_i - E(x) ]² * P(x_i) = Σ [ x_i - μ_x ]² * P(x_i)
• Normal random variable = z-score = z = (X - μ)/σ
• Chi-square statistic = Χ² = [ ( n - 1 ) * s² ] / σ²
• f statistic = f = [ s₁²/σ₁² ] / [ s₂²/σ₂² ]
• Expected value of sum of random variables =
  E(X + Y) = E(X) + E(Y)
• Expected value of difference between random variables =
  E(X - Y) = E(X) - E(Y)
• Variance of the sum of independent random variables =
  Var(X + Y) = Var(X) + Var(Y)
• Variance of the difference between independent random variables =
  Var(X - Y) = E(X) + E(Y)

Sampling Distributions

• Mean of sampling distribution of the mean = μ_x = μ
• Mean of sampling distribution of the proportion = μ_p = P
• Standard deviation of proportion = σ_p = sqrt[ P * (1 - P)/n ] = sqrt( PQ / n )
• Standard deviation of the mean = σ_x = σ/sqrt(n)
• Standard deviation of difference of sample means =
  σ_d = sqrt[ (σ₁² / n₁) + (σ₂² / n₂) ]
• Standard deviation of difference of sample proportions =
  σ_d = sqrt{ [P₁(1 - P₁) / n₁] + [P₂(1 - P₂) / n₂] }

Standard Error

• Standard error of proportion = SE_p = s_p = sqrt[ p * (1 - p)/n ] = sqrt( pq / n )
• Standard error of difference for proportions =
  SE_p = s_p = sqrt{ p * ( 1 - p ) * [ (1/n₁) + (1/n₂) ] }
• Standard error of the mean = SE_x = s_x = s/sqrt(n)
• Standard error of difference of sample means =
  SE_d = s_d = sqrt[ (s₁² / n₁) + (s₂² / n₂) ]
• Standard error of difference of paired sample means =
  SE_d = s_d = { sqrt [ (Σ(d_i - d)² / (n - 1) ] } / sqrt(n)
• Pooled sample standard error = s_pooled = sqrt [ (n₁ - 1) * s₁² + (n₂ - 1) * s₂² ] / (n₁ + n₂ - 2) ]
• Standard error of difference of sample proportions =
  s_d = sqrt{ [p₁(1 - p₁) / n₁] + [p₂(1 - p₂) / n₂] }

Discrete Probability Distributions

• Binomial formula: P(X = x) = b(x; n, P) = _nC_x * P^x * (1 - P)^{n - x} = _nC_x * P^x * Q^{n - x}
• Mean of binomial distribution = μ_x = n * P
• Variance of binomial distribution = σ_x² = n * P * ( 1 - P )
• Negative Binomial formula: P(X = x) = b*(x; r, P) = _x-1C_r-1 * P^r * (1 - P)^{x - r}
• Mean of negative binomial distribution = μ_x = rQ / P
• Variance of negative binomial distribution = σ_x² = r * Q / P²
• Geometric formula: P(X = x) = g(x; P) = P * Q^{x - 1}
• Mean of geometric distribution = μ_x = Q / P
• Variance of geometric distribution = σ_x² = Q / P²
• Hypergeometric formula: P(X = x) = h(x; N, n, k) = [ _kC_x ] [ _N-kC_n-x ] / [ _NC_n ]
• Mean of hypergeometric distribution = μ_x = n * k / N
• Variance of hypergeometric distribution = σ_x² = n * k * ( N - k ) * ( N - n ) / [ N² * ( N - 1 ) ]
• Poisson formula: P(x; μ) = (e^-μ) (μ^x) / x!
• Mean of Poisson distribution = μ_x = μ
• Variance of Poisson distribution = σ_x² = μ
• Multinomial formula: P = [ n! / ( n₁! * n₂! * ... n_k! ) ] * ( p₁^n₁ * p₂^n₂ * . . . * p_k^n_k )

Linear Transformations

For the following formulas, assume that Y is a linear transformation of the random variable X, defined by the equation: Y = aX + b.

• Mean of a linear transformation = E(Y) = Y = aX + b.
• Variance of a linear transformation = Var(Y) = a² * Var(X).
• Standardized score = z = (x - μ_x) / σ_x.
• t-score = t = (x - μ_x) / [ s/sqrt(n) ].

Estimation

• Confidence interval: Sample statistic + Critical value * Standard error of statistic
• Margin of error = (Critical value) * (Standard deviation of statistic)
• Margin of error = (Critical value) * (Standard error of statistic)

Hypothesis Testing

• Standardized test statistic = (Statistic - Parameter) / (Standard deviation of statistic)
• One-sample z-test for proportions: z-score = z = (p - P₀) / sqrt( p * q / n )
• Two-sample z-test for proportions: z-score = z = z = [ (p₁ - p₂) - d ] / SE
• One-sample t-test for means: t-score = t = (x - μ) / SE
• Two-sample t-test for means: t-score = t = [ (x₁ - x₂) - d ] / SE
• Matched-sample t-test for means: t-score = t = [ (x₁ - x₂) - D ] / SE = (d - D) / SE
• Chi-square test statistic = Χ² = Σ[ (Observed - Expected)² / Expected ]

Degrees of Freedom

The correct formula for degrees of freedom (DF) depends on the situation (the nature of the test statistic, the number of samples, underlying assumptions, etc.).

• One-sample t-test: DF = n - 1
• Two-sample t-test: DF = (s₁²/n₁ + s₂²/n₂)² / { [ (s₁² / n₁)² / (n₁ - 1) ] + [ (s₂² / n₂)² / (n₂ - 1) ] }
• Two-sample t-test, pooled standard error: DF = n₁ + n₂ - 2
• Simple linear regression, test slope: DF = n - 2
• Chi-square goodness of fit test: DF = k - 1
• Chi-square test for homogeneity: DF = (r - 1) * (c - 1)
• Chi-square test for independence: DF = (r - 1) * (c - 1)

Sample Size

Below, the first two formulas find the smallest sample sizes required to achieve a fixed margin of error, using simple random sampling. The third formula assigns sample to strata, based on a proportionate design. The fourth formula, Neyman allocation, uses stratified sampling to minimize variance, given a fixed sample size. And the last formula, optimum allocation, uses stratified sampling to minimize variance, given a fixed budget.

• Mean (simple random sampling): n = { z² * σ² * [ N / (N - 1) ] } / { ME² + [ z² * σ² / (N - 1) ] }
• Proportion (simple random sampling): n = [ ( z² * p * q ) + ME² ] / [ ME² + z² * p * q / N ]
• Proportionate stratified sampling: n_h = ( N_h / N ) * n
• Neyman allocation (stratified sampling): n_h = n * ( N_h * σ_h ) / [ Σ ( N_i * σ_i ) ]
• Optimum allocation (stratified sampling):
  n_h = n * [ ( N_h * σ_h ) / sqrt( c_h ) ] / [ Σ ( N_i * σ_i ) / sqrt( c_i ) ]

Normal Distribution and Scales

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