Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

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278 CHAPTER 9 | Introduction to the t Statisticprecisely, two events (or observations) are independent if the occurrence of the firstevent has no effect on the probability of the second event. We examined specificexamples of independence and non-independence in Box 8.1 (page 244).2. The population sampled must be normal.This assumption is a necessary part of the mathematics underlying the developmentof the t statistic and the t distribution table. However, violating this assumptionhas little practical effect on the results obtained for a t statistic, especiallywhen the sample size is relatively large. With very small samples, a normal populationdistribution is important. With larger samples, this assumption can be violatedwithout affecting the validity of the hypothesis test. If you have reason to suspectthat the population distribution is not normal, use a large sample to be safe.■ The Influence of Sample Size and Sample VarianceAs we noted in Chapter 8 (page 242), a variety of factors can influence the outcome of ahypothesis test. In particular, the number of scores in the sample and the magnitude of thesample variance both have a large effect on the t statistic and thereby influence the statisticaldecision. The structure of the t formula makes these factors easier to understand.t 5 M 2ms Mwhere s M5Îs2nBecause the estimated standard error, s M, appears in the denominator of the formula, alarger value for s Mproduces a smaller value (closer to zero) for t. Thus, any factor thatinfluences the standard error also affects the likelihood of rejecting the null hypothesis andfinding a significant treatment effect. The two factors that determine the size of the standarderror are the sample variance, s 2 , and the sample size, n.The estimated standard error is directly related to the sample variance so that the largerthe variance, the larger the error. Thus, large variance means that you are less likely toobtain a significant treatment effect. In general, large variance is bad for inferential statistics.Large variance means that the scores are widely scattered, which makes it difficultto see any consistent patterns or trends in the data. In general, high variance reduces thelikelihood of rejecting the null hypothesis.On the other hand, the estimated standard error is inversely related to the number ofscores in the sample. The larger the sample, the smaller the error. If all other factors areheld constant, large samples tend to produce bigger t statistics and therefore are more likelyto produce significant results. For example, a 2-point mean difference with a sample ofn = 4 may not be convincing evidence of a treatment effect. However, the same 2-pointdifference with a sample of n = 100 is much more compelling.LEARNING CHECK1. If a t statistic is computed for a sample of n = 4 scores with SS = 300, then whatare the sample variance and the estimated standard error for the sample mean?a. s 2 = 10 and s M= 5b. s 2 = 100 and s M= 20c. s 2 = 10 and s M= 20d. s 2 = 100 and s M= 5
SECTION 9.3 | Measuring Effect Size for the t Statistic 2792. A sample is selected from a population with μ = 40, and a treatment is administeredto the sample. If the sample variance is s 2 = 96, which set of sample characteristicshas the greatest likelihood of rejecting the null hypothesis?a. M = 37 for a sample of n = 16b. M = 37 for a sample of n = 64c. M = 34 for a sample of n = 16d. M = 34 for a sample of n = 643. A sample is selected from a population and a treatment is administered to thesample. If there is a 2-point difference between the sample mean and the originalpopulation mean, which set of sample characteristics has the greatest likelihood ofrejecting the null hypothesis?a. s 2 = 12 for a sample with n = 25b. s 2 = 12 for a sample with n = 9c. s 2 = 32 for a sample with n = 25d. s 2 = 32 for a sample with n = 9ANSWERS1. D, 2. D, 3. A9.3 Measuring Effect Size for the t StatisticLEARNING OBJECTIVES5. Calculate Cohen’s d or the percentage of variance accounted for (r 2 ) to measureeffect size for a hypothesis test with a t statistic.6. Explain how measures of effect size for a t test are influenced by sample size andsample variance.7. Explain how a confidence interval can be used to describe the size of a treatmenteffect and describe the factors that affect with width of a confidence interval.8. Describe how the results from a hypothesis test using a t statistic are reported in theliterature.In Chapter 8 we noted that one criticism of a hypothesis test is that it does not really evaluatethe size of the treatment effect. Instead, a hypothesis test simply determines whetherthe treatment effect is greater than chance, where “chance” is measured by the standarderror. In particular, it is possible for a very small treatment effect to be “statistically significant,”especially when the sample size is very large. To correct for this problem, it isrecommended that the results from a hypothesis test be accompanied by a report of effectsize such as Cohen’s d.■ Estimated Cohen’s dWhen Cohen’s d was originally introduced (page 251), the formula was presented asCohen’s d 5mean differencestandard deviation 5 m treatment 2 m no treatments
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EDITION10Statistics for theBehavior
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Statistics for the Behavioral Scien
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CONTENTSCHAPTER 1 Introduction to S
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CONTENTSvii5.4 Using z-Scores to St
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CONTENTSixCHAPTER 10 The t Test for
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CONTENTSxiCHAPTER 15 Correlation 48
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PREFACEMany students in the behavio
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PREFACExv3. The former Chapter 19,
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PREFACExviiTo the StudentA primary
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ABOUT THE AUTHORSFREDERICK J GRAVET
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PREVIEWBefore we begin our discussi
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4 CHAPTER 1 | Introduction to Stati
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10 CHAPTER 1 | Introduction to Stat
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PREVIEWThere is some evidence that
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36 CHAPTER 2 | Frequency Distributi
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Central TendencyCHAPTER3Tools You W
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SECTION 3.1 | Overview 69DEFINITION
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SECTION 3.2 | The Mean 71research r
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SECTION 3.2 | The Mean 73the distri
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SECTION 3.2 | The Mean 75TABLE 3.1S
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SECTION 3.2 | The Mean 77Table 3.2
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SECTION 3.3 | The Median 793.3 The
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SECTION 3.3 | The Median 81To find
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SECTION 3.4 | The Mode 832. What is
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SECTION 3.4 | The Mode 85FIGURE 3.7
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SECTION 3.5 | Selecting a Measure o
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SECTION 3.6 | Central Tendency and
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FOCUS ON PROBLEM SOLVING 95of centr
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PROBLEMS 976. A population of N = 1
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VariabilityCHAPTER4Tools You Will N
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SECTION 4.1 | Introduction to Varia
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SECTION 4.2 | Defining Standard Dev
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SECTION 4.3 | Measuring Variance an
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SECTION 4.4 | Measuring Standard De
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SECTION 4.5 | Sample Variance as an
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SECTION 4.6 | More about Variance a
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SUMMARY 1252. A population has a me
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FOCUS ON PROBLEM SOLVING 127VAR0000
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PROBLEMS 1299. For the following se
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z-Scores: Location of Scoresand Sta
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SECTION 5.1 | Introduction to z-Sco
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SECTION 5.2 | z-Scores and Location
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SECTION 5.2 | z-Scores and Location
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SECTION 5.3 | Other Relationships B
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SECTION 5.4 | Using z-Scores to Sta
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SECTION 5.4 | Using z-Scores to Sta
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SECTION 5.5 | Other Standardized Di
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SECTION 5.5 | Other Standardized Di
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SECTION 5.6 | Computing z-Scores fo
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SECTION 5.7 | Looking Ahead to Infe
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SUMMARY 153LEARNING CHECK1. In N =
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DEMONSTRATION 5.2 155sure that your
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PROBLEMS 15716. A distribution of e
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PREVIEWHow likely is it that you wi
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162 CHAPTER 6 | Probabilitythe prob
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164 CHAPTER 6 | Probability(Note: W
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166 CHAPTER 6 | ProbabilityFIGURE 6
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168 CHAPTER 6 | Probability■ The
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170 CHAPTER 6 | ProbabilityFinding
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172 CHAPTER 6 | Probabilityand exac
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174 CHAPTER 6 | ProbabilityFinally,
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176 CHAPTER 6 | ProbabilityXz-score
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178 CHAPTER 6 | ProbabilityBOX 6.1
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180 CHAPTER 6 | ProbabilityEXAMPLE
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182 CHAPTER 6 | ProbabilityFIGURE 6
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184 CHAPTER 6 | Probability6.5 Look
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186 CHAPTER 6 | Probability2. For a
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188 CHAPTER 6 | ProbabilityDEMONSTR
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190 CHAPTER 6 | Probability6. Draw
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Probability and Samples: TheDistrib
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SECTION 7.1 | Samples, Populations,
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SECTION 7.1 | Samples, Populations,
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SECTION 7.2 | The Distribution of S
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SECTION 7.3 | Probability and the D
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SECTION 7.3 | Probability and the D
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SECTION 7.4 | More about Standard E
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SECTION 7.4 | More about Standard E
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FOCUS ON PROBLEM SOLVING 219SUMMARY
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PROBLEMS 221STEP 2Compute the z-sco
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Introduction toHypothesis TestingCH
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SECTION 8.1 | The Logic of Hypothes
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Page 263 and 264: SECTION 8.3 | More about Hypothesis
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Page 347 and 348: KEY TERMS 325SUMMARY1. The independ
Page 349 and 350: FOCUS ON PROBLEM SOLVING 327Group S
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PROBLEMS 329The t Statistic Finally
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PROBLEMS 331c. Write a sentence dem
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PROBLEMS 33319. If other factors ar
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PREVIEWIt’s the night before an e
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338 CHAPTER 11 | The t Test for Two
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PREVIEWYour job interview is tomorr
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368 CHAPTER 12 | Introduction to An
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Repeated-MeasuresAnalysis of Varian
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SECTION 13.1 | Overview of the Repe
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SECTION 13.2 | Hypothesis Testing a
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SECTION 13.3 | More about the Repea
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SPSS ® 4375. When the obtained F-r
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DEMONSTRATION 13.1 439you must also
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PROBLEMS 441PROBLEMS1. How does the
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PROBLEMS 44313. The following summa
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PROBLEMS 44523. The following data
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Two-FactorAnalysis of Variance(Inde
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SECTION 14.1 | An Overview of the T
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SECTION 14.2 | An Example of the Tw
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SECTION 14.3 | More about the Two-F
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SUMMARY 473SUMMARY1. A research stu
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FOCUS ON PROBLEM SOLVING 475Descrip
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DEMONSTRATION 14.1 477STEP 2STAGE 1
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PROBLEMS 479PROBLEMS1. Define each
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PROBLEMS 48115. Research results in
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PROBLEMS 483EasyFactorA TaskDifficu
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PREVIEWA recent report describing t
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488 CHAPTER 15 | CorrelationDEFINIT
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492 CHAPTER 15 | CorrelationSubstit
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494 CHAPTER 15 | Correlationeither
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496 CHAPTER 15 | Correlationreliabl
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498 CHAPTER 15 | Correlation60Numbe
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500 CHAPTER 15 | CorrelationOne of
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502 CHAPTER 15 | CorrelationBOX 15.
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504 CHAPTER 15 | Correlation171615Z
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506 CHAPTER 15 | Correlation3. What
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508 CHAPTER 15 | Correlationon the
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510 CHAPTER 15 | CorrelationLEARNIN
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512 CHAPTER 15 | CorrelationScoresR
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514 CHAPTER 15 | CorrelationThe pro
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516 CHAPTER 15 | Correlationthat a
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518 CHAPTER 15 | Correlationmeasure
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520 CHAPTER 15 | CorrelationSUMMARY
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522 CHAPTER 15 | CorrelationTo comp
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524 CHAPTER 15 | CorrelationSTEP 2C
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526 CHAPTER 15 | Correlation14. Ide
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Introduction to RegressionCHAPTER16
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SECTION 16.1 | Introduction to Line
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SECTION 16.2 | The Standard Error o
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SECTION 16.3 | Introduction to Mult
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KEY TERMS 553a predicted portion an
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DEMONSTRATION 16.1 555DEMONSTRATION
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PROBLEMS 557Alzheimer’s disease.
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The Chi-Square Statistic:Tests for
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SECTION 17.1 | Introduction to Chi-
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SECTION 17.2 | An Example of the Ch
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SECTION 17.3 | The Chi-Square Test
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SECTION 17.4 | Effect Size and Assu
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SECTION 17.4 | Effect Size and Assu
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SECTION 17.5 | Special Applications
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SECTION 17.5 | Special Applications
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SUMMARY 591With df = 2 and α = .05
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SPSS ® SPSS ® 593General instruct
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DEMONSTRATION 17.1 595FOCUS ON PROB
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PROBLEMS 597Finally, we can add the
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PROBLEMS 599response, a sample of 1
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PROBLEMS 601a researcher obtains a
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PREVIEWIn 1960, Gibson and Walk des
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606 CHAPTER 18 | The Binomial Testp
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608 CHAPTER 18 | The Binomial Test2
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610 CHAPTER 18 | The Binomial Test
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612 CHAPTER 18 | The Binomial Test
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614 CHAPTER 18 | The Binomial TestT
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616 CHAPTER 18 | The Binomial Testt
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618 CHAPTER 18 | The Binomial TestB
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Basic Mathematics ReviewAPPENDIXAPR
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APPENDIX A | Basic Mathematics Revi
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648 APPENDIX B | Statistical Tables
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664 APPENDIX C | Solutions for Odd-
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684 APPENDIX D | General Instructio
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Hypothesis Tests for OrdinalData: M
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APPENDIX E | Hypothesis Tests for O
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702 Statistics Organizer: Finding t
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ReferencesAckerman, R. (2011). Meta
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REFERENCES 719Alzheimer’s disease
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REFERENCES 721of aging and estrogen
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724 NAME INDEXKaell, A., 622Kahn, K
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726 SUBJECT INDEXConfidence interva
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728 SUBJECT INDEXIndependent random
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730 SUBJECT INDEXResearch studies,
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732 SUBJECT INDEXTwo-factor ANOVA (
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Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

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