TABLES AND FORMULAS FOR MOORE Basic Practice of Statistics

TABLES AND FORMULAS FOR MOORE Basic Practice of Statistics

Exploring Data: Distributions

? Look for overall pattern (shape, center, spread) and deviations (outliers).

? Mean (use a calculator):

x = x1 + x2 + ? ? ? + xn = 1

n

n

xi

? Standard deviation (use a calculator):

s=

1 n-1

(xi - x)2

? Median: Arrange all observations from smallest to largest. The median M is located (n + 1)/2 observations from the beginning of this list.

? Quartiles: The first quartile Q1 is the median of the observations whose position in the ordered list is to the left of the location of the overall median. The third quartile Q3 is the median of the observations to the right of the location of the overall median.

? Five-number summary:

Minimum, Q1, M, Q3, Maximum

? Standardized value of x: z= x-?

Exploring Data: Relationships

? Look for overall pattern (form, direction, strength) and deviations (outliers, influential observations).

? Correlation (use a calculator):

r

=

1 n-1

xi - x sx

yi - y sy

? Least-squares regression line (use a calculator): y^ = a + bx with slope b = rsy/sx and intercept a = y - bx

? Residuals:

residual = observed y - predicted y = y - y^

Producing Data

? Simple random sample: Choose an SRS by giving every individual in the population a numerical label and using Table B of random digits to choose the sample.

? Randomized comparative experiments:

Random ?? B Group 1 E Treatment 1 rrj Observe Allocationrr j Group 2 E Treatment 2 ??B Response

Probability and Sampling Distributions

? Probability rules:

? Any probability satisfies 0 P (A) 1. ? The sample space S has probability

P (S) = 1. ? If events A and B are disjoint, P (A or B) =

P (A) + P (B). ? For any event A, P (A does not occur) =

1 - P (A)

? Sampling distribution of a sample mean: ? x has mean ? and standard deviation /n.

? x has a Normal distribution if the population distribution is Normal.

? Central limit theorem: x is approximately Normal when n is large.

Basics of Inference

? z confidence interval for a population mean ( known, SRS from Normal population):

x ? z n

z from N (0, 1)

? Sample size for desired margin of error m: n = z 2 m

? z test statistic for H0 : ? = ?0 ( known, SRS from Normal population):

z = x -?0 / n

P -values from N (0, 1)

Inference About Means

? t confidence interval for a population mean (SRS from Normal population):

x ? t s n

t from t(n - 1)

? t test statistic for H0 : ? = ?0 (SRS from Normal population):

t = x -?0 s/ n

P -values from t(n - 1)

? Matched pairs: To compare the responses to the two treatments, apply the one-sample t procedures to the observed differences.

? Two-sample t confidence interval for ?1 - ?2 (independent SRSs from Normal populations):

(x1 - x2) ? t

s21 + s22 n1 n2

with conservative t from t with df the smaller of n1 - 1 and n2 - 1 (or use software).

? Two-sample t test statistic for H0 : ?1 = ?2 (independent SRSs from Normal populations):

t = x1 - x2 s21 + s22 n1 n2

with conservative P -values from t with df the smaller of n1 - 1 and n2 - 1 (or use software).

Inference About Proportions

? Sampling distribution of a sample proportion: when the population and the sample size are both large and p is not close to 0 or 1, p^ is approximately Normal with mean p and standard deviation p(1 - p)/n.

? Large-sample z confidence interval for p:

p^ ? z p^(1 - p^) n

z from N (0, 1)

Plus four to greatly improve accuracy: use the same formula after adding 2 successes and two failures to the data.

? z test statistic for H0 : p = p0 (large SRS):

z = p^ - p0 p0(1 - p0) n

P -values from N (0, 1)

? Sample size for desired margin of error m:

n=

z

2

p(1 - p)

m

where p is a guessed value for p or p = 0.5.

? Large-sample z confidence interval for p1 - p2:

(p^1 - p^2) ? zSE

z from N (0, 1)

where the standard error of p^1 - p^2 is

SE = p^1(1 - p^1) + p^2(1 - p^2)

n1

n2

Plus four to greatly improve accuracy: use the same formulas after adding one success and one failure to each sample.

? Two-sample z test statistic for H0 : p1 = p2 (large independent SRSs):

z=

p^1 - p^2

p^(1 - p^) 1 + 1

n1 n2

where p^ is the pooled proportion of successes.

The Chi-Square Test

? Expected count for a cell in a two-way table:

expected

count

=

row

total ? column table total

total

? Chi-square test statistic for testing whether the row and column variables in an r ? c table are unrelated (expected cell counts not too small):

X2 =

(observed count - expected count)2 expected count

with P -values from the chi-square distribution with df = (r - 1) ? (c - 1).

? Describe the relationship using percents, comparison of observed with expected counts, and terms of X2.

Inference for Regression

? Conditions for regression inference: n observations on x and y. The response y for any fixed x has a Normal distribution with mean given by the true regression line ?y = + x and standard deviation . Parameters are , , .

? Estimate by the intercept a and by the slope b of the least-squares line. Estimate by the regression standard error:

s=

1 n-2

residual2

Use software for all standard errors in regression.

? t confidence interval for regression slope :

b ? tSEb

t from t(n - 2)

? t test statistic for no linear relationship, H0 : = 0:

t

=

b SEb

P -values from t(n - 2)

? t confidence interval for mean response ?y when x = x:

y^ ? tSE?^

t from t(n - 2)

? t prediction interval for an individual observation y when x = x:

y^ ? tSEy^

t from t(n - 2)

One-way Analysis of Variance: Comparing Several Means

? ANOVA F tests whether all of I populations have the same mean, based on independent SRSs from I Normal populations with the same . P -values come from the F distribution with I -1 and N - I degrees of freedom, where N is the total observations in all samples.

? Describe the data using the I sample means and standard deviations and side-by-side graphs of the samples.

? The ANOVA F test statistic (use software) is F = MSG/MSE, where

MSG

=

n1(x1 - x)2 + ? ? ? + nI (xI - x)2 I -1

MSE = (n1 - 1)s21 + ? ? ? + (nI - 1)s2I N -I

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TABLES

Table A Standard Normal Probabilities Table B Random Digits Table C t Distribution Critical Values Table D Chi-square Distribution Critical Values Table E Critical Values of the Correlation r

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TABLES

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Table entry for z is the area under the standard Normal curve to the left of z.

Table entry

z

T A B L E A STANDARD NORMAL CUMULATIVE PROPORTIONS

z

3.4 3.3 3.2 3.1 3.0

2.9 2.8 2.7 2.6 2.5

2.4 2.3 2.2 2.1 2.0

1.9 1.8 1.7 1.6 1.5

1.4 1.3 1.2 1.1 1.0

0.9 0.8 0.7 0.6 0.5

0.4 0.3 0.2 0.1 0.0

.00

.0003 .0005 .0007 .0010 .0013

.0019 .0026 .0035 .0047 .0062

.0082 .0107 .0139 .0179 .0228

.0287 .0359 .0446 .0548 .0668

.0808 .0968 .1151 .1357 .1587

.1841 .2119 .2420 .2743 .3085

.3446 .3821 .4207 .4602 .5000

.01

.0003 .0005 .0007 .0009 .0013

.0018 .0025 .0034 .0045 .0060

.0080 .0104 .0136 .0174 .0222

.0281 .0351 .0436 .0537 .0655

.0793 .0951 .1131 .1335 .1562

.1814 .2090 .2389 .2709 .3050

.3409 .3783 .4168 .4562 .4960

.02

.0003 .0005 .0006 .0009 .0013

.0018 .0024 .0033 .0044 .0059

.0078 .0102 .0132 .0170 .0217

.0274 .0344 .0427 .0526 .0643

.0778 .0934 .1112 .1314 .1539

.1788 .2061 .2358 .2676 .3015

.3372 .3745 .4129 .4522 .4920

.03

.0003 .0004 .0006 .0009 .0012

.0017 .0023 .0032 .0043 .0057

.0075 .0099 .0129 .0166 .0212

.0268 .0336 .0418 .0516 .0630

.0764 .0918 .1093 .1292 .1515

.1762 .2033 .2327 .2643 .2981

.3336 .3707 .4090 .4483 .4880

.04

.0003 .0004 .0006 .0008 .0012

.0016 .0023 .0031 .0041 .0055

.0073 .0096 .0125 .0162 .0207

.0262 .0329 .0409 .0505 .0618

.0749 .0901 .1075 .1271 .1492

.1736 .2005 .2296 .2611 .2946

.3300 .3669 .4052 .4443 .4840

.05

.0003 .0004 .0006 .0008 .0011

.0016 .0022 .0030 .0040 .0054

.0071 .0094 .0122 .0158 .0202

.0256 .0322 .0401 .0495 .0606

.0735 .0885 .1056 .1251 .1469

.1711 .1977 .2266 .2578 .2912

.3264 .3632 .4013 .4404 .4801

.06

.0003 .0004 .0006 .0008 .0011

.0015 .0021 .0029 .0039 .0052

.0069 .0091 .0119 .0154 .0197

.0250 .0314 .0392 .0485 .0594

.0721 .0869 .1038 .1230 .1446

.1685 .1949 .2236 .2546 .2877

.3228 .3594 .3974 .4364 .4761

.07

.0003 .0004 .0005 .0008 .0011

.0015 .0021 .0028 .0038 .0051

.0068 .0089 .0116 .0150 .0192

.0244 .0307 .0384 .0475 .0582

.0708 .0853 .1020 .1210 .1423

.1660 .1922 .2206 .2514 .2843

.3192 .3557 .3936 .4325 .4721

.08

.0003 .0004 .0005 .0007 .0010

.0014 .0020 .0027 .0037 .0049

.0066 .0087 .0113 .0146 .0188

.0239 .0301 .0375 .0465 .0571

.0694 .0838 .1003 .1190 .1401

.1635 .1894 .2177 .2483 .2810

.3156 .3520 .3897 .4286 .4681

.09

.0002 .0003 .0005 .0007 .0010

.0014 .0019 .0026 .0036 .0048

.0064 .0084 .0110 .0143 .0183

.0233 .0294 .0367 .0455 .0559

.0681 .0823 .0985 .1170 .1379

.1611 .1867 .2148 .2451 .2776

.3121 .3483 .3859 .4247 .4641

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