Download A Statistics Summary Cheat Sheet and more Cheat Sheet Statistics in PDF only on Docsity! A Statistics Summary-sheet Sampling Conditions Confidence Interval Test Statistic σ 2 is known ⇒ X ∼ N (µ, σ2/n) ± n ZX σ α 2/ n X Z / 0 σ µ− = Yes σ 2 is unknown ⇒ X ∼ N (µ, σ2/n) ± n s ZX 2/α ns X Z / 0 µ− = Is n is large, say over 30? − ± n pp Zp )1( 2/α n pp pp Z )1( 00 0 − − = No X ∼ N (µ, σ 2 ) and σ 2 is known ⇒ X ∼ N (µ, σ2/n) ± n ZX σ α 2/ n X Z / 0 σ µ− = X ∼ N (µ, σ 2 ) and σ 2 is unknown ⇒ X ∼ tn-1 (µ, σ 2 /n) ± − n s tX n 2/,1 α ns X t n / 0 1 µ− = − If n is not large, say over 30 and X is not ∼ N (µ, σ 2 ), cannot proceed with parametric statistics. Formulas, Distributions, and Concepts Counting and Probabilities )!xn( !n Pxn − = Permutations )!xn(!x !n Cxn − = Combinations )B(P )BA(P )B|A(P ∩ = Conditional Probability )B(P)B|A(P)BA(P =∩ Probability of an Intersection Discrete Probability Distributions xnx x )p1(p )!xn(!x !n )x(P − − − = Binomial Probability ! )( x e xP x x µµ− = Poisson Probability Continuous Probability Distributions Random Variable ∼ Distribution (mean, variance) Standard Normal Z ∼ N(0,1) Estimating Sample Size 2 22 /2 E z n σα= For estimation and CI for the population mean, normal population, σ2 known, or estimated by a pilot run. E = absolute error. Hypothesis Testing 1. Set up the appropriate null which must be in equality form, always and alternative hypotheses. 2. Define the rejection area. Take care as to whether the test is one-tailed or two-tailed. Look to the alternative hypothesis to determine this. 3. Calculate the test statistic. 4. State Decision. 5. Interpret your conclusion. Hypothesis Testing (test statistics and their distributions under the null) n -X 0 σ µ ∼ αz When population variance known, or if n ≥30, substitute s for σ. n -X 0 s µ ∼ tn-1,α When If population is normal, population variance unknown. ( ) + −−− 21 2 021 11 )( nn s YX µµ ∼ α,221 t −+nn where 2 )1()1( 21 2 22 2 112 −+ −+− = nn snsn s Difference in means, independent samples, population variances unknown, but statistically almost equal. n/)1( p-p 00 0 pp − ∼ Zα Population Proportion, n ≥30 Formulas for ANOVA j jn i ij j n x X ∑ == 1 ( ) 1 1 2 2 − − = ∑ = j jn i jij j n Xx s T jn i ij k j n x X ∑∑ == = 11 Overall Sample Mean or if the treatment sample sizes are all equal, k X X k j j∑ = = 1 1− = k SSTR MSTR ( )∑ = −= k j jj XXnSSTR 1 2 kn SSE MSE T − = ∑ = −= k j jj snSSE 1 2)1( MSE MSTR F = ∼ kTnkF −− ,1 Terms and Concepts Central Limit Theorem: If the sample size n is large, say n ≥30 no matter what the population distribution is, the sampling distribution of the sample mean tends towards the normal as n gets large.
