Statistics I
1. Basic statistics
        Parameters (for populations)    m, s², s
        Statistics (for samples)              x, S², S,

        Variance S²
        Standard deviation S
        Normal distribution
        Significance a

        Parametric statistics
                for test distributions with known parameters
        Non-parametric statistics
                parameters are unknown
                non-normal distributions, small sample sizes
                use low rank data such as nominal and ordinal
        Parametric is more powerful when the parameters are known
        Otherwise non-parametric is more powerful

2. t test
Test for equality of means of two samples
Assumptions: random samples, normal distribution, equal variance
Null hypothesis: h₀: X₁ = X₂

            X₁-X₂                   1      1                (n₁-1)S₁² + (n₂-1)S₂²
        t = ------,   S_e = S_p  --- + ---,    S_p² = --------------------
                S_e                    n₁     n₂                   (n₁ -1) + (n₂ - 1)

Compare the computed t value to the t table value (two-tailed) for
specified degrees of freedom and level of significance

If the t > +critical value or t < -critical value, reject the H₀
otherwise accept the null hypothesis that the two means are from the same
population.

3. Mann-Whitney Test
Nonparametric substitute for T test of the equality of two means
Null hypothesis:
Combine the two sets (n,m) of data and rank them from 1 to n+m

               n                n(n + 1)
        T = S R(X_i) - ----------,  R(X_i) R(Y_i) are the ranks of X_i, Y_i
               1                      2

Compare the computed T value to the T table values (two-tailed)
for specified sample size (n) and level of significance

For the upper critical value
        T_1-a = nm - T_a
Tied data are assigned averaged ranks, e.g. R(X_i)=R(Y_i)=(8+9)/2=8.5

4. C² Test
Test for goodness of fit between a sample and a predefined distribution
can be used for nominal and ordinal data, i.e. count data
can be used for nonparametric statistics

Null hypothesis: the sample has a normal distribution

                 k   (O_j - E_j)²
        X² = S -------------'      O_j- number of observed
                 1         E_j                E_j- number of expected

standardize the data:
                 X_i - X
        Z_i = --------
                    S
Divide the normal distribution evenly into n categories
assign the sample into the same n categories
Compare the computed C² value to the C² table values (one-tailed) for
specified degrees of freedom and level of significance

If X² value > critical value, reject the null hypothesis
otherwise accept the null that the sample has a normal distribution

5. Kolmogorov-Smirnov Test
Nonparametric substitute for X² test
It does not group data into categories
It is more sensitive to deviations in the tails

Fit a sample to a normal distribution of unspecified m and s
Null hypothesis:
standardize the data
                  Xi - X
        Zi = ----------
                        S
plot a normal distribution and the sample in cumulative form
find the maximum absolute difference between the two curves

        K-S = |normal - sample|

compare the computed K-S value to K-S table values (one/two-tailed)
for specified sample size and level of significance

If the K-S value > critical value, reject the null hypothesis
 

Statistics II  Regression Analysis
1. Joint variation of two variables
        Joint variation of two variables about their common mean
        Covariance

2. Simple regression and least square methods
        Regression: model relationships between variables
        Y_i = b₀ + b₁X_i + e_i,   b₀ - intercept  b_i - slope
                                              n
        Least square methods:    S (Y_i - Y_i)² = minimum; Se_i = 0
                                              1
        Parameter estimates: b₀, b

        Yi = b₀ + bX_i

3. Goodness of fit (coefficient of determination)
                                                           n
        Total Sum of Squares:      SS_t =  S (Y_i - Y)²
                                                           1
                                                             n
        Sum Squares of regression: SS_r =  S(Y_i - Y)²
                                                             1
                                                            n
        Sum Squares of residuals:  SS_e =  S (Y_i - Y_i)²
                                                            1
        SS_t = SS_r + SS_e

        Coefficient of determination (goodness of fit): R² = SS_r/SS_t

        Coefficient of correlation: R = R² = SS_r/SS_t
                                               r = Cov(x,y)/s_xs_y

                                                   (k-1)(1 - R²)
        adjusted R²: R²a = R² -    -------------------
                                                        N - k
        N - sample size, k - number of independent variables

4. Test of regression model
General F test: equality of two variances
        Null hypothesis: S₁² = S₂²
                S₁²
        F = ------
                S₂²
        Compare the computed F value to the F table values for  specified
degrees of freedom for both vairiances and level of significance

        If the computed F>critical F, reject the null, accept otherwise

F test for regression model:
        Null hypothesis: SS_r = SS_e
                   SS_r/k
        F = ----------,     k - number of parameters excluding b₀
             SS_e/N-k-1     N - sample size

t test for individual parameters b
        Null hypothesis: bi = 0
                  bi
        t = ------,  S_bi - standard error of bi
                S_bi

5. Multiple regression
        Yi = b₀ + b₁X₁ + b₂X₂ + b₃X₃ + ... + bmXm + ei

        Yi = b₀ + b₁X₁ + b₂X₂ + b₃X₃ + ... + bmXm
 
6. Other regressions 
 
 

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