TABLES AND FORMULAS FOR MOORE Basic Practice of Statistics Exploring Data: Distributions Correlation (use a calculator): • Look for overall pattern (shape, center, spread) 1 x x y y • r = i − i − and deviations (outliers). n 1 sx % sy & − ! # $ Mean (use a calculator): • Least-squares regression line (use a calculator): • x1 + x2 + + xn 1 yˆ = a + bx with slope b = rsy/sx and intercept x = ··· = xi a = y bx n n − ! Residuals: Standard deviation (use a calculator): • • residual = observed y predicted y = y yˆ − − 1 2 s = (xi x) "n 1 − − ! Producing Data Median: Arrange all observations from smallest • Simple random sample: Choose an SRS by to largest. The median M is located (n + 1)/2 • observations from the beginning of this list. giving every individual in the population a numerical label and using Table B of random Quartiles: The first quartile Q is the median of digits to choose the sample. • 1 the observations whose position in the ordered Randomized comparative experiments: list is to the left of the location of the overall • median. The third quartile Q is the median of 3 Group 1 - Treatment 1 the observations to the right of the location of Random ¨¨* HHj Observe the overall median. AllocationHHj ¨¨* Response Group 2 - Treatment 2 Five-number summary: • Minimum,Q1, M, Q3, Maximum Probability and Sampling Distributions Standardized value of x: • Probability rules: x µ • z = − Any probability satisfies 0 P (A) 1. σ • ≤ ≤ The sample space S has probability • P (S) = 1. Exploring Data: Relationships If events A and B are disjoint, P (A or B)= • P (A)+P (B). Look for overall pattern (form, direction, • For any event A, P (A does not occur) = strength) and deviations (outliers, influential • 1 P (A) observations). − Sampling distribution of a sample mean: Two-sample t test statistic for H : µ = µ • • 0 1 2 (independent SRSs from Normal populations): x has mean µ and standard deviation σ/√n. • x has a Normal distribution if the popula- x1 x2 • t = − tion distribution is Normal. s2 s2 1 + 2 Central limit theorem: x is approximately "n1 n2 • Normal when n is large. with conservative P -values from t with df the smaller of n1 1 and n2 1 (or use software). Basics of Inference − − z confidence interval for a population mean Inference About Proportions • (σ known, SRS from Normal population): Sampling distribution of a sample proportion: σ • x z∗ z∗ from N(0, 1) when the population and the sample size are ± √n both large and p is not close to 0 or 1,p ˆ is ap- Sample size for desired margin of error m: proximately Normal with mean p and standard • deviation p(1 p)/n. z σ 2 − n = ∗ ' m Large-sample z confidence interval for p: # $ • z test statistic for H0 : µ = µ0 (σ known, SRS pˆ(1 pˆ) • pˆ z∗ − z∗ from N(0, 1) from Normal population): ± " n x µ z = − 0 P -values from N(0, 1) Plus four to greatly improve accuracy: use the σ/√n same formula after adding 2 successes and two failures to the data. z test statistic for H : p = p (large SRS): Inference About Means • 0 0 t confidence interval for a population mean (SRS pˆ p0 • z = − P -values from N(0, 1) from Normal population): p (1 p ) 0 − 0 s " n x t∗ t∗ from t(n 1) ± √n − Sample size for desired margin of error m: t test statistic for H0 : µ = µ0 (SRS from Normal • • 2 population): z∗ n = p∗(1 p∗) m x µ # $ − t = − 0 P -values from t(n 1) s/√n − where p∗ is a guessed value for p or p∗ =0.5. Matched pairs: To compare the responses to the Large-sample z confidence interval for p1 p2: • • − two treatments, apply the one-sample t proce- (ˆp pˆ ) z∗SE z∗ from N(0, 1) dures to the observed differences. 1 − 2 ± Two-sample t confidence interval for µ µ (in- where the standard error ofp ˆ1 pˆ2 is • 1 − 2 − dependent SRSs from Normal populations): pˆ1(1 pˆ1) pˆ2(1 pˆ2) 2 2 SE = − + − s1 s2 " n1 n2 (x x ) t∗ + 1 2 n n − ± " 1 2 Plus four to greatly improve accuracy: use the with conservative t∗ from t with df the smaller same formulas after adding one success and one of n 1 and n 1 (or use software). failure to each sample. 1 − 2 − Two-sample z test statistic for H : p = p t confidence interval for regression slope β: • 0 1 2 • (large independent SRSs): b t∗SEb t∗ from t(n 2) pˆ pˆ ± − z = 1 − 2 1 1 t test statistic for no linear relationship, pˆ(1 pˆ) + • n n H0 : β = 0: " − # 1 2 $ b wherep ˆ is the pooled proportion of successes. t = P -values from t(n 2) SEb − The Chi-Square Test t confidence interval for mean response µ when • y x = x∗: Expected count for a cell in a two-way table: • row total column total yˆ t∗SEµˆ t∗ from t(n 2) expected count = × ± − table total t prediction interval for an individual observa- • Chi-square test statistic for testing whether the tion y when x = x∗: • row and column variables in an r c table are × yˆ t∗SE t∗ from t(n 2) unrelated (expected cell counts not too small): ± yˆ − (observed count expected count)2 X2 = − expected count ! One-way Analysis of Variance: with P -values from the chi-square distribution Comparing Several Means with df = (r 1) (c 1). − × − ANOVA F tests whether all of I populations Describe the relationship using percents, com- • • have the same mean, based on independent SRSs parison of observed with expected counts, and from I Normal populations with the same σ. terms of X2. P -values come from the F distribution with I 1 − and N I degrees of freedom, where N is the − Inference for Regression total observations in all samples. Conditions for regression inference: n observa- Describe the data using the I sample means and • • tions on x and y. The response y for any fixed standard deviations and side-by-side graphs of x has a Normal distribution with mean given by the samples. the true regression line µ = α + βx and stan- y The ANOVA F test statistic (use software) is dard deviation σ. Parameters are α, β, σ. • F = MSG/MSE, where Estimate α by the intercept a and β by the slope • n (x x)2 + + n (x x)2 b of the least-squares line. Estimate σ by the MSG = 1 1 − ··· I I − I 1 regression standard error: − (n 1)s2 + +(n 1)s2 MSE = 1 − 1 ··· I − I 1 N I s = residual2 − "n 2 − ! Use software for all standard errors in regression. P1: OSO FREE013-TABLE FREE013-Moore September 4, 2008 7:30 690 TABLES Table entry for z is the area under the standard Normal curve to the Table entry left of z. z TABLE A Standard Normal cumulative proportions z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 −3.4 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0002 −3.3 .0005 .0005 .0005 .0004 .0004 .0004 .0004 .0004 .0004 .0003 −3.2 .0007 .0007 .0006 .0006 .0006 .0006 .0006 .0005 .0005 .0005 −3.1 .0010 .0009 .0009 .0009 .0008 .0008 .0008 .0008 .0007 .0007 −3.0 .0013 .0013 .0013 .0012 .0012 .0011 .0011 .0011 .0010 .0010 −2.9 .0019 .0018 .0018 .0017 .0016 .0016 .0015 .0015 .0014 .0014 −2.8 .0026 .0025 .0024 .0023 .0023 .0022 .0021 .0021 .0020 .0019 −2.7 .0035 .0034 .0033 .0032 .0031 .0030 .0029 .0028 .0027 .0026 −2.6 .0047 .0045 .0044 .0043 .0041 .0040 .0039 .0038 .0037 .0036 −2.5 .0062 .0060 .0059 .0057 .0055 .0054 .0052 .0051 .0049 .0048 −2.4 .0082 .0080 .0078 .0075 .0073 .0071 .0069 .0068 .0066 .0064 −2.3 .0107 .0104 .0102 .0099 .0096 .0094 .0091 .0089 .0087 .0084 −2.2 .0139 .0136 .0132 .0129 .0125 .0122 .0119 .0116 .0113 .0110 −2.1 .0179 .0174 .0170 .0166 .0162 .0158 .0154 .0150 .0146 .0143 −2.0 .0228 .0222 .0217 .0212 .0207 .0202 .0197 .0192 .0188 .0183 −1.9 .0287 .0281 .0274 .0268 .0262 .0256 .0250 .0244 .0239 .0233 −1.8 .0359 .0351 .0344 .0336 .0329 .0322 .0314 .0307 .0301 .0294 −1.7 .0446 .0436 .0427 .0418 .0409 .0401 .0392 .0384 .0375 .0367 −1.6 .0548 .0537 .0526 .0516 .0505 .0495 .0485 .0475 .0465 .0455 −1.5 .0668 .0655 .0643 .0630 .0618 .0606 .0594 .0582 .0571 .0559 −1.4 .0808 .0793 .0778 .0764 .0749 .0735 .0721 .0708 .0694 .0681 −1.3 .0968 .0951 .0934 .0918 .0901 .0885 .0869 .0853 .0838 .0823 −1.2 .1151 .1131 .1112 .1093 .1075 .1056 .1038 .1020 .1003 .0985 −1.1 .1357 .1335 .1314 .1292 .1271 .1251 .1230 .1210 .1190 .1170 −1.0 .1587 .1562 .1539 .1515 .1492 .1469 .1446 .1423 .1401 .1379 −0.9 .1841 .1814 .1788 .1762 .1736 .1711 .1685 .1660 .1635 .1611 −0.8 .2119 .2090 .2061 .2033 .2005 .1977 .1949 .1922 .1894 .1867 −0.7 .2420 .2389 .2358 .2327 .2296 .2266 .2236 .2206 .2177 .2148 −0.6 .2743 .2709 .2676 .2643 .2611 .2578 .2546 .2514 .2483 .2451 −0.5 .3085 .3050 .3015 .2981 .2946 .2912 .2877 .2843 .2810 .2776 −0.4 .3446 .3409 .3372 .3336 .3300 .3264 .3228 .3192 .3156 .3121 −0.3 .3821 .3783 .3745 .3707 .3669 .3632 .3594 .3557 .3520 .3483 −0.2 .4207 .4168 .4129 .4090 .4052 .4013 .3974 .3936 .3897 .3859 −0.1 .4602 .4562 .4522 .4483 .4443 .4404 .4364 .4325 .4286 .4247 −0.0 .5000 .4960 .4920 .4880 .4840 .4801 .4761 .4721 .4681 .4641 Revised Pages P1: OSO FREE013-TABLE FREE013-Moore September 4, 2008 7:30 TABLES 691 Table entry Table entry for z is the area under the standard Normal curve to the left of z.