TABLES AND FORMULAS FOR MOORE Basic Practice of Statistics

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Virgil Dean
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1 TABLES AND FORMULAS FOR MOORE Basic Practice of Statistics Explorig Data: Distributios Look for overall patter (shape, ceter, spread) ad deviatios (outliers). Mea (use a calculator): x = x 1 + x x = 1 xi Stadard deviatio (use a calculator): 1 s = (xi x) 1 2 Media: Arrage all observatios from smallest to largest. The media M is located ( + 1)/2 observatios from the begiig of this list. Quartiles: The first quartile Q 1 is the media of the observatios whose positio i the ordered list is to the left of the locatio of the overall media. The third quartile Q 3 is the media of the observatios to the right of the locatio of the overall media. Five-umber summary: Miimum, Q 1, M, Q 3, Maximum Stadardized value of x: z = x µ σ Explorig Data: Relatioships Look for overall patter (form, directio, stregth) ad deviatios (outliers, ifluetial observatios). Correlatio (use a calculator): r = 1 ( ) x i x ( ) y i y 1 Least-squares regressio lie (use a calculator): ŷ = a + bx with slope b = rs y /s x ad itercept a = y bx Residuals: residual = observed y predicted y = y ŷ Producig Data Simple radom sample: Choose a SRS by givig every idividual i the populatio a umerical label ad usig Table B of radom digits to choose the sample. s x Radomized comparative experimets: Group 1 Treatmet 1 Radom Observe Allocatio Group 2 Respose Treatmet 2 Probability ad Samplig Distributios Probability rules: Ay probability satisfies 0 P (A) 1. The sample space S has probability P (S) = 1. For ay evet A, P (A does ot occur) = 1 P (A) If evets A ad B are disjoit, P (A or B) = P (A)+P(B). s y

2 Samplig distributio of a sample mea: x has mea µ ad stadard deviatio σ/. x has a Normal distributio if the populatio distributio is Normal. Cetral limit theorem: x is approximately Normal whe is large. Basics of Iferece z cofidece iterval for a populatio mea (σ kow, SRS from Normal populatio): x ± z σ z from N(0, 1) Sample size for desired margi of error m: ( z ) σ 2 = m z test statistic for H 0 : µ = µ 0 (σ kow, SRS from Normal populatio): z = x µ 0 σ/ P -values from N(0, 1) Iferece About Meas t cofidece iterval for a populatio mea (SRS from Normal populatio): x ± t s t from t( 1) t test statistic for H 0 : µ = µ 0 (SRS from Normal populatio): t = x µ 0 s/ P -values from t( 1) Matched pairs: To compare the resposes to the two treatmets, apply the oe-sample t procedures to the observed differeces. Two-sample t cofidece iterval for µ 1 µ 2 (idepedet SRSs from Normal populatios): (x 1 x 2 ) ± t s s with coservative t from t with df the smaller of 1 1ad 2 1 (or use software). Two-sample t test statistic for H 0 : µ 1 = µ 2 (idepedet SRSs from Normal populatios): t = x 1 x 2 s s with coservative P -values from t with df the smaller of 1 1ad 2 1 (or use software). Iferece About Proportios Samplig distributio of a sample proportio: whe the populatio ad the sample size are both large ad p is ot close to 0 or 1, ˆp is approximately Normal with mea p ad stadard deviatio p(1 p)/. Large-sample z cofidece iterval for p: ˆp ± z ˆp(1 ˆp) z from N(0, 1) Plus four to greatly improve accuracy: use the same formula after addig 2 successes ad two failures to the data. z test statistic for H 0 : p = p 0 (large SRS): ˆp p 0 z = p 0 (1 p 0 ) P -values from N(0, 1) Sample size for desired margi of error m: ( z ) 2 = p (1 p ) m where p is a guessed value for p or p =0.5. Large-sample z cofidece iterval for p 1 p 2 : (ˆp 1 ˆp 2 ) ± z SE z from N(0, 1) where the stadard error of ˆp 1 ˆp 2 is ˆp 1 (1 ˆp 1 ) SE = + ˆp 2(1 ˆp 2 ) 1 2 Plus four to greatly improve accuracy: use the same formulas after addig oe success ad oe failure to each sample.

3 Two-sample z test statistic for H 0 : p 1 = p 2 (large idepedet SRSs): ˆp 1 ˆp 2 z = ( 1 ˆp(1 ˆp) + 1 ) 1 2 where ˆp is the pooled proportio of successes. The Chi-Square Test Expected cout for a cell i a two-way table: expected cout = row total colum total table total Chi-square test statistic for testig whether the row ad colum variables i a r c table are urelated (expected cell couts ot too small): X 2 = (observed cout expected cout) 2 expected cout with P -values from the chi-square distributio with df = (r 1) (c 1). Describe the relatioship usig percets, compariso of observed with expected couts, ad terms of X 2. Iferece for Regressio The regressio model: We have observatios o x ad y. The respose y for ay fixed x has a Normal distributio with mea give by the true regressio lie µ y = α + βx ad stadard deviatio σ. Parameters are α, β, σ. Estimate α by the itercept a ad β by the slope b of the least-squares lie. Estimate σ by the regressio stadard error: 1 s = residual 2 2 t cofidece iterval for regressio slope β: b ± t SE b t from t( 2) t test statistic for o liear relatioship, H 0 : β =0: t = b SE b P -values from t( 2) t cofidece iterval for mea respose µ y whe x = x : ŷ ± t SEˆµ t from t( 2) t predictio iterval for a idividual observatio y whe x = x : ŷ ± t SEŷ t from t( 2) Oe-way Aalysis of Variace: Comparig Several Meas ANOVA F tests whether all of I populatios have the same mea, based o idepedet SRSs from I Normal populatios with the same σ. P - values come from the F distributio with I 1 ad N I degrees of freedom, where N is the total observatios i all samples. Describe the data usig the I sample meas ad stadard deviatios ad side-by-side graphs of the samples. The ANOVA F test statistic (use software) is F =MSG/MSE, where MSG = 1(x 1 x) I (x I x) 2 I 1 MSE = ( 1 1)s ( I 1)s 2 I N I Use software for all stadard errors i regressio.

4 TABLE A Stadard Normal probabilities z

5 TABLE B Radom digits Lie TABLE C t distributio critical values Upper tail probability p df z % 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8% 99.9% Cofidece level C

6 TABLE E Chi-square distributio critical values Upper tail probability p df TABLE F Critical values of the correlatio r Upper tail probability p

TABLES AND FORMULAS FOR MOORE Basic Practice of Statistics

TABLES AND FORMULAS FOR MOORE Basic Practice of Statistics Explorig Data: Distributios Look for overall patter (shape, ceter, spread) ad deviatios (outliers). Mea (use a calculator): x = x 1 + x 2 + +