Schaum'S Outline Of Theory And Problems Of Beginning Statistics

CONTENTSChapter 10INFERENCES FOR TWO POPULATIONS.vii211x,Sampling Distribution offor Large Independent Samples.Estimation of p1- p2 Using Large Independent Samples. TestingHypothesis about p I- p, Using Large Independent Samples.-x2Sampling Distribution of XIfor Small Independent Samplesfrom Normal Populations with Equal (but unknown) StandardDeviations. Estimation of p 1- p, Using Small Independent Samples froinNormal Populations with Equal (but unknown) Standard Deviations.Testing Hypothesis about p,- p, Using Small IndependentSamples from Normal Populations with Equal (but Unknown) StandardDeviations. Sampling Distribution of XIfor Small Independent Samplesfrom Normal Populations with Unequal (and Unknown) Standard Deviations.Estimation of p, - p2 Using Small Independent Samples from Normalx2Populations with Unequal (and Unknown) Standard Deviations. TestingHypothesis about p 1- p2 Using Small Independent Samples from NormalPopulations with Unequal (and Unknown) Standard Deviations.Sampling Distribution of for Normally Distributed DifferencesComputed from Dependent Samples. Estimation of pd Using NormallyaDistributed Differences Computed from Dependent Samples. TestingHypothesis about pd Using Normally Distributed Differences Computedfrom Dependent Samples. Sampling Distribution of PI- p2 for Large IndependentSamples. Estimation of PI - P2 Using Large Independent Samples. TestingHypothesis about PI- P2 Using Large Independent Samplcs.Chapter 11CHI-SQUARE PROCEDURES.249Chi-square Distribution. Chi-square Tables. Goodness-of-Fit Test.Observed and Expected Frequencies. Sampling Distribution of theGoodness-of-Fit Test Statistic. Chi-square Independence Test.Sampling Distribution of the Test Statistic for the Chi-squareIndependence Test. Sampling Distribution of the Sample Variance.Inferences Concerning the Population Variance.Chapter 12ANALYSIS OF VARIANCE (ANOVA) .F Distribution. F Table. Logic Behind a One-way ANOVA. Sum ofSquares, Mean Squares, and Degrees of Freedom for a One-way ANOVA.Sampling Distribution for the One-way ANOVA Test Statistic. BuildingOne-way ANOVA Tables and Testing the Equality of Means. LogicBehind a Two-way ANOVA. Sum of Squares, Mean Squares, and Degreesof Freedom for a Two-way ANOVA. Building Two-way ANOVA Tables.Sampling Distributions for the Two-way ANOVA. Testing HypothesisConcerning Main Effects and Interaction.272

- .CONTENTSVlllChapter 13REGRESSION AND CORRELATION.309Straight Lines. Linear Regression Model. Least Squares Line. Error Sumof Squares. Standard Deviation of Errors. Total Sum of Squares.Regression Sum of Squares. Coefficient of Determination. hiean, StandardDeviaticjii, and Sampling Distribution of the Slope of the Estimated RegressionEquation. Inferences Concerning the Slope of the Population Regression Line.Estimation and Prediction in Linear Regression. Linear CorrelationCoefficient. Inference Concerning the Population Correlation Coefficient.Chapter 14NONPARAMETRIC STATISTICS.334NonparametricMethods. Sign Test. Wilcoxon Signed-Ranks Test for TwoDependent Samples. Wilcoxon Rank-Sum Test for Two Independent Samples.Kruskal-Wall6 Test. Rank Correlation. Runs Test for Randomness.APPENDIX - . 1Binomial Probabilities .3592Areas under the Standard Normal Curve from 0 to 2 .3643Area in the Right Tail under the t Distribution Curve .3654Area in the Right Tail under the Chi-square Distribution Curve .3665Area in the Right Tail under the F Distribution Curve .367INDEX .-369

Chapter 1IntroductionSTATISTICSStatistics is a discipline of study dealing with the collection, analysis, interpretation, andpresentation of data. Statistical methodology is utilized by pollsters who sample our opinionsconcerning topics ranging from art to zoology. Statistical methodology is also utilized by businessand industry to help control the quality of goods and services that they produce. Social scientists andpsychologists use statistical methodology to study our behaviors. Because of its broad range ofapplicability, a course in statistics is required of majors in disciplines such as sociology, psychology,criminal justice, nursing, exercise science, pharmacy, education, and many others. To accommodatethis diverse group of users, examples and problems in this outline are chosen from many differentsources.DESCRIPTIVE STATISTICSThe use of graphs, charts, and tables and the calculation of various statistical measures toorganize and summarize information is called descriptive statistics. Descriptive statistics help toreduce our information to a manageable size and put it into focus.EXAMPLE 1.1 The compilation of batting average, runs batted in, runs scored, and number of home runs foreach player, as well as earned run average, wordlost percentage, number of saves, etc., for each pitcher from theofficial score sheets for major league baseball players is an example of descriptive statistics. These statisticalmeasures allow us to compare players, determine whether a player is having an “off year” or “good year,” etc.EXAMPLE 1.2 The publication entitled Crime in the United States published by the Federal Bureau ofInvestigation gives summary information concerning various crimes for the United States. The statisticalmeasures given in this publication are also examples of descriptive statistics and they are useful to individuals inlaw enforcement.INFERENTIAL STATISTICS: POPULATION AND SAMPLEThe complete collection of individuals, items, or data under consideration in a statistical studyis referred to as the pupulatiurz. The portion of the population selected for analysis is called thesample. Inferential statistics consists of techniques for reaching conclusions about a populationbased upon information contained in a sample.EXAMPLE 1.3 The results of polls are widely reported by both the written and the electronic media. Thetechniques of inferential statistics are widely utilized by pollsters. Table 1.1 gives several examples ofpopulations and samples encountered in polls reported by the media. The methods of inferential statistics areused to make inferences about the populations based upon the results found in the samples and to give anindication about the reliability of these inferences. The results of a poll of 600 registered voters might bereported as follows: Forty percent of the voters approve of the president’s economic policies. The margin oferror for the survey is 4%. The survey indicates that an estimated 40% of all registered voters approve of theeconomic policies, but it might be as low as 36% or as high as 44%.1

2INTRODUCTION[CHAP. 1Table 1.1PopulationAll registered votersSampleA telephone survey of 600 registered votersAll owners of handgunsA telephone survey of 1000 handgun ownersHouseholds headed by a single parentThe results from questionnaires sent to 2500households headed by a single parentThe CEOs of all private companiesThe results from surveys sent to 150 CEO’s ofprivate companiesEXAMPLE 1.4 The techniques of inferential statistics are applied in many industrial processes to control thequality of the products produced. In industrial settings, the population may consist of the daily production oftoothbrushes, computer chips, bolts, and so forth. The sample will consist of a random and representativeselection of items from the process producing the toothbrushes, computer chips, bolts, etc. The informationcontained in the daily samples is used to construct control charts. The control charts are then used to monitor thequality of the products.EXAMPLE 1.5 The statistical methods of inferential statistics are used to analyze the data collected inresearch studies. Table 1.2 gives the samples and populations for several such studies. The informationcontained in the samples is utilized to make inferences concerning the populations. If it is found that 245 of 350or 70% of prison inmates in a criminal justice study were abused as children, what conclusions may be inferredconcerning the percent of all prison inmates who were abused as children? The answers to this question arefound in Chapters 8 and 9.PopulationAll prison inmatesSampleA criminal justice study of 350 prison inmatesLegal aliens living in the United StatesA sociological study conducted by a universityresearcher of 200 legal aliensAlzheimer patients in the United StatesA medical study of 75 such patientsconducted by a university hospitalAdult children of alcoholicsA psychological study of 200 such individualsVARIABLE, OBSERVATION, AND DATA SETA characteristic of interest concerning the individual elements of a population or a sample iscalled a variable. A variable is often represented by a letter such as x, y, or z. The value of a variablefor one particular element from the sample or population is called an Observation. A data set consistsof the observations of a variable for the elements of a sample.EXAMPLE 1.6 Six hundred registered voters are polled and each one is asked if they approve or disapproveof the president’s economic policies. The variable is the registered voter’s opinion of the president’s economicpolicies. The data set consists of 600 observations. Each observation will be the response “approve” or theresponse “do not approve.’’ If the response “approve” is coded as the number 1 and the response “do notapprove’’ is coded as 0, then the data set will consist of 600 observations, each one of which is either 0 or 1. If xis used t o represent the variable, then x can assume two values, 0 or I .

CHAP. 11INTRODUCTION3EXAMPLE 1.7 A survey of 2500 households headed by a single parent is conducted and one characteristic ofinterest is the yearly household income. The data set consists of the 2500 yearly household incomes for theindividuals in the survey. If y is used to represent the variable, then the values for y will be between the smallestand the largest yearly household incomes for the 2500 households.EXAMPLE 1.8 The number of speeding tickets issued by 75 Nebraska state troopers for the month of June isrecorded. The data set consists of 75 observations.QUANTITATIVE VARIABLE: DISCRETE AND CONTINUOUS VARIABLEA quantitative variable is determined when the description of the characteristic of interest resultsin a numerical value. When a measurement is required to describe the characteristic of interest or it isnecessary to perform a count to describe the characteristic, a quantitative variable is defined. Adiscrete variable is a quantitative variable whose values are countable. Discrete variables usuallyresult from counting. A continuous variable is a quantitative variable that can assume any numericalvalue over an interval or over several intervals. A continuous variable usually results from making ameasurement of some type.EXAMPLE 1.9 Table 1.3 gives several discrete variables and the set of possible values for each one. In eachcase the value of the variable is determined by counting. For a given box of 100 diabetic syringes, the number ofdefective needles is determined by counting how many of the 100 are defective. The number of defectives foundmust equal one of the 101 values listed. The number of possible outcomes is finite for each of the first fourvariables; that is, the number of possible outcomes are 101, 31, 501, and 5 1 respectively. The number ofpossible outcomes for the last variable is infinite. Since the number of possible outcomes is infinite andcountable for this variable, we say that the number of outcomes is countably infinite.Sometimes it is not clear whether a variable is discrete or continuous. Test scores expressed as apercent, for example, are usually given as whole numbers between 0 and 100. It is possible to give ascore such as 75.57565. However, this is not done in practice because teachers are unable to evaluateto this degree of accuracy. This variable is usually regarded as continuous, although for all practicalpurposes, it is discrete. To summarize, due to measurement limitations, many continuous variablesactually assume only a countable number of values.Table 1.3Discrete variableThe number of defective needles in boxes of 100 diabeticsyringesPossible values for the variable0, I , 2 , . . . , 100The number of individuals in groups of 30 with a type Apersonality0 , 1 , 2, . . . , 30The number of surveys returned out of 500 mailed insociological studies0, 1,2, . . . , 500The number of prison inmates in 50 having finished highschool or obtained a GED who are selected for criminaljustice studies0 , 1 , 2, . . . , 50The number of times you need to flip a coin before a headappears for the first time1 , 2 , 3 , . .(there is no upper limit since conceivablyone might need to flip forever to obtain thefirst head),

INTRODUCTION4[CHAP. 1EXAMPLE 1.10 Table 1.4 gives several continuous variables and the set of possible values for each one. Allthree continuous variables given in Table 1.4 involve measurement, whereas the variables in Example I .9 allinvolve counting.Table 1.4Possible values for the variableContinuous variableThe length of prison time served forAll the real numbers between a and b,where a is the smallest amount of timeindividuals convicted of first degree murderserved and b is the largest amountThe household income for households withincomes less than or equal to 20,000All the real numbers between a and 20,000, where a is the smallesthousehold income in the populationThe cholesterol reading for thoseindividuals having cholesterol readingsequal to or greater than 200 mg/dlAll real numbers between 200 and b,where b is the largest cholesterol readingof all such individualsQUALITATIVE VARIABLEA qualitative variable is determined when the description of the characteristic of interest resultsin a nonnumerical value. A qualitative variable may be classified into two or more categories.EXAMPLE 1.11 Table 1.5 gives several examples of qualitative variables along with a set of categories intowhich they may be classified.Qualitative variableMarital statusPossible categories for the variableSingle, married, divorced, separatedGenderMale, femaleCrime classificationMisdemeanor, felonyPain levelNone, low, moderate, severePersonality typeType A, type BThe possible categories for qualitative variables are often coded for the purpose of performingcomputerized statistical analysis. Marital status might be coded as 1, 2, 3, or 4, where 1 representssingle, 2 represents married, 3 represents divorced, and 4 represents separated. The variable gendermight be coded as 0 for female and 1 for male. The categories for any qualitative variable may becoded in a similar fashion. Even though numerical values are associated with the characteristic ofinterest after being coded, the variable is considered a qualitative variable.NOMINAL, ORDINAL, INTERVAL, AND RATIO LEVELS OF MEASUREMENTThere are four levels of meusuremerit or scales of measurements into which data can beclassified. The nominal scale applies to data that are used for category identification. The nominallevel of measi rernertt is characterized by data that consist of names, labels, or categories only.Nominal scale duta cannot be arranged in an ordering scheme. The arithmetic operations of addition,subtraction, multiplication, and division are not performed for nominal data.

5INTRODUCTIONCHAP. 13EXAMPLE 1.12 Table 1.6 gives several qualitative variables and a set of possible nominal level data values.The data values are often encoded for recording in a computer data tile. Blood type might be recorded as 1, 2, 3,or 4; state of residence might be recorded as 1,2, . . . , or 50; and type of crime might be recorded as 0 or I , or 1or 2, etc. Similarly, color of road sign could be recorded as 1, 2, 3, 4, or 5 and religion could be recorded as 1,2, or 3. There is no order associated with these data and arithmetic operations are not performed. For example,adding Christian and Moslem ( 1 2) does not give other (3).Qualitative variableBlood typePossible nominal level data valuesassociated with the variableA, B, AB, 0State of residenceAlabama,. . . , WyomingType of crimeMisdemeanor, felonyColor of road signs in the state of NebraskaRed, white, blue, brown, greenReligionChristian, Moslem, otherThe ordinal scale applies to data that can be arranged in some order, but differences betweendata values either cannot be determined or are meaningless. The ordinal level of measurement ischaracterized by data that applies to categories that can be ranked. Ordinal scale data can bearranged in an ordering scheme.EXAMPLE 1.13 Table 1.7 gives several qualitative variables and a set of possible ordinal level data values.The data values for ordinal level data are often encoded for inclusion in computer data files. Arithmeticoperations are not performed on ordinal level data, but an ordering scheme exists. A full-size automobile islarger than a subcompact, a tire rated excellent is better than one rated poor, no pain is preferable to any levelof pain, the level of play in major league baseball is better than the level of play in class AA, and so forth.Qualitative variableAutomobile size descriptionPossible ordinal level data values associatedwith the variableSubcompact, compact, intermediate, full-sizeProduct ratingPoor, good, excellentSocioeconomic classLower, middle, upperPain levelNone, low, moderate, severeBaseball team classificationClass A, class AA, class AAA , major leagueThe interval scale applies to data that can be arranged in some order and for which differences indata values are meaningful. The interval level of measurement results from counting or measuring.Interval scale data can be arranged in an ordering scheme and differences can be calculated andinterpreted. The value zero is arbitrarily chosen for interval data and does not imply an absence ofthe characteristic being measured. Ratios are not meaningful for interval data.EXAMPLE 1.14 Stanford-Binet IQ scores represent interval level data. Joe’s IQ score equals 100 and John’sIQ score equals 150. John has a higher IQ than Joe; that is, IQ scores can be arranged in order. John’s IQ scoreis 50 points higher than Joe’s IQ score; that is, differences can be calculated and interpreted. However, wecannot conclude that John is 1.5 ti

Probability Distribution. Mean of a Discrete Random Variable. Standard Deviation of a Discrete Random Variable. Binomial Random Variable. Binomial Probability Formula. Tables of the Binomial Distribution. Mean and Standard Deviation of a Binomial Random Variable. Poisson Random Variable. Poisson Probability Formula. Hypergeome tric Random Variable.