TABLES AND FORMULAS FOR MOORE Basic Practice of Statistics
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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. s y The sample space S has probability P (S) = 1. If evets A ad B are disjoit, P (A or B) = P (A)+P (B). For ay evet A, P (A does ot occur) = 1 P (A)
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 1 ad 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 1 ad 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 Coditios for regressio iferece: 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 BMTables.idd Page /15/11 4:25:16 PM user-s163 Tables Table A Stadard Normal Probabilities Table B Radom Digits Table C t Distributio Critical Values Table D Chi-square Distributio Critical Values Table E Critical Values of the Correlatio r 675
5 BMTables.idd Page /15/11 4:25:16 PM user-s TABLES Table etry for z is the area uder the stadard Normal curve to the left of z. Table etry z TABLE A Stadard Normal cumulative proportios z
6 BMTables.idd Page /15/11 4:25:16 PM user-s163 TABLES 677 Table etry for z is the area uder the stadard Normal curve to the left of z. Table etry z TABLE A Stadard Normal cumulative proportios (cotiued) z
7 BMTables.idd Page /15/11 4:25:16 PM user-s TABLES TABLE B Radom digits LINE
8 BMTables.idd Page /15/11 4:25:16 PM user-s163 Table etry for C is the critical value t* required for cofidece level C. To approximate oe- ad two-sided P-values, compare the value of the t statistic with the critical values of t* that match the P-values give at the bottom of the table. Area C t* t* Tail area 1 C 2 TABLE C t distributio critical values CONFIDENCE LEVEL C DEGREES OF FREEDOM 50% 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8% 99.9% z* Oe-sided P Two-sided P
9 BMTables.idd Page /15/11 4:25:17 PM user-s TABLES Table etry for p is the critical value x* with probability p lyig to its right. Probability p χ* TABLE D Chi-square distributio critical values df p
10 BMTables.idd Page /15/11 4:25:17 PM user-s163 TABLES 681 Table etry for p is the critical value r* of the correlatio coefficiet r with probability p lyig to its right. Probability p r* TABLE E Critical values of the correlatio r UPPER TAIL PROBABILITY p
11 FrotEdpapers.idd Page 2 11/15/11 3:01:53 PM user-s163 STATISTICS IN SUMMARY Plot your data: Stemplot, histogram Aalyzig Data for Oe Variable Iterpret what you see: Shape, ceter, spread, outliers Numerical summary? x ad s, five-umber summary? Desity curve? Normal distributio? STATISTICS IN SUMMARY Plot your data: Scatterplot Aalyzig Data for Two Variables Iterpret what you see: Directio, form, stregth. Liear? Numerical summary? x, y, s x, s y, ad r? Regressio lie? STATISTICS IN SUMMARY Oe sample Mea (quatitative data) Chap. 18 t = x µ s/ Test a claim Sigificace test Two samples Proportio (categorical data) Compare meas (quatitative data) Chap. 20 Chap. 19 z = ˆp p0 p0(1 p0) ( x1 x2) (µ1 µ2) t = s s2 2 2 State problem Compare proportios (categorical data) Chap. 21 z = ˆp 1 ˆp 2 ˆp(1 ˆp)( ) ˆp = pooled proportio Mea (quatitative data) Chap. 18 x ± t s Oe sample Estimate a parameter Cofidece iterval Proportio (categorical data) Differece of meas (quatitative data) Chap. 20 Chap. 19 ˆp ± z ˆp(1 ˆp) ( x 1 x 2) ± t s s2 2 2 (use plus four) Two samples Differece of proportios (categorical data) Chap. 21 (ˆp 1 ˆp 2) ± z ˆp 1(1 ˆp 1) ˆp2(1 ˆp2) (use plus four)
12 FrotEdpapers.idd Page 3 11/15/11 3:01:53 PM user-s163 ORGANIZING A STATISTICAL PROBLEM: A Four-Step Process STATE: What is the practical questio, i the cotext of the real-world settig? PLAN: What specific statistical operatios does this problem call for? SOLVE: Make the graphs ad carry out the calculatios eeded for this problem. CONCLUDE: Give your practical coclusio i the settig of the real-world problem. CONFIDENCE INTERVALS: The Four-Step Process STATE: What is the practical questio that requires estimatig a parameter? PLAN: Idetify the parameter, choose a level of cofidece, ad select the type of cofidece iterval that fits your situatio. SOLVE: Carry out the work i two phases: 1. Check the coditios for the iterval you pla to use. 2. Calculate the cofidece iterval. CONCLUDE: Retur to the practical questio to describe your results i this settig. TESTS OF SIGNIFICANCE: A Four-Step Process STATE: What is the practical questio that requires a statistical test? PLAN: Idetify the parameter, state ull ad alterative hypotheses, ad choose the type of test that fits your situatio. SOLVE: Carry out the test i three phases: 1. Check the coditios for the test you pla to use. 2. Calculate the test statistic. 3. Fid the P-value. CONCLUDE: Retur to the practical questio to describe your results i this settig.
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