PowerPoint Presentation by Charlie Cook The University of West Alabama William G. Zikmund Barry J. Babin 9 th Edition Part 6 Data Analysis and Presentation © 2007 Thomson/South-Western. All rights reserved. Chapter 21 Univariate Statistical Analysis

© 2007 Thomson/South-Western. All rights reserved.21–2 LEARNING OUTCOMES 1.Implement the hypothesis-testing procedure. 2.Use p-values to assess statistical significance. 3.Test a hypotheses about an observed mean compared to some standard. 4.Know the difference between Type I and Type II errors. 5.Know when a univariate χ 2 test is appropriate and how to conduct one. After studying this chapter, you should be able to

© 2007 Thomson/South-Western. All rights reserved.21–3 Hypothesis Testing Types of HypothesesTypes of Hypotheses  Relational hypotheses  Examine how changes in one variable vary with changes in another.  Hypotheses about differences between groups  Examine how some variable varies from one group to another.  Hypotheses about differences from some standard  Examine how some variable differs from some preconceived standard. These tests typify univariate statistical tests.

© 2007 Thomson/South-Western. All rights reserved.21–4 Statistical Analysis: Key Terms HypothesisHypothesis  Unproven proposition: a supposition that tentatively explains certain facts or phenomena  An assumption about nature of the world. Null HypothesisNull Hypothesis  Statement about the status quo  No difference in sample and population Alternative HypothesisAlternative Hypothesis  Statement that indicates the opposite of the null hypothesis

© 2007 Thomson/South-Western. All rights reserved.21–5 Types of Statistical Analysis Univariate Statistical AnalysisUnivariate Statistical Analysis  Tests of hypotheses involving only one variable.  Testing of statistical significance Bivariate Statistical AnalysisBivariate Statistical Analysis  Tests of hypotheses involving two variables. Multivariate Statistical AnalysisMultivariate Statistical Analysis  Statistical analysis involving three or more variables or sets of variables.

© 2007 Thomson/South-Western. All rights reserved.21–6 The Hypothesis-Testing Procedure ProcessProcess  The specifically stated hypothesis is derived from the research objectives.  A sample is obtained and the relevant variable is measured.  The measured sample value is compared to the value either stated explicitly or implied in the hypothesis.  If the value is consistent with the hypothesis, the hypothesis is supported.  If the value is not consistent with the hypothesis, the hypothesis is not supported.

© 2007 Thomson/South-Western. All rights reserved.21–7 Significance Levels and p-Values Significance LevelSignificance Level  A critical probability associated with a statistical hypothesis test that indicates how likely an inference supporting a difference between an observed value and some statistical expectation is true.  The acceptable level of Type I error. p-Valuep-Value  Probability value, or the observed or computed significance level; p-values are compared to significance levels to test hypotheses.  Higher p-values equal more support for an hypothesis.

© 2007 Thomson/South-Western. All rights reserved.21–8 EXHIBIT 21.1 p -Values and Statistical Tests

© 2007 Thomson/South-Western. All rights reserved.21–9 EXHIBIT 21.1 p -Values and Statistical Tests (cont’d)

© 2007 Thomson/South-Western. All rights reserved.21–10 EXHIBIT 21.2 P-Values As the observed mean gets further from the standard (proposed population mean), the p-value decreases. The lower the p-value, the more confidence you have that the sample mean is different.

© 2007 Thomson/South-Western. All rights reserved.21–11 An Example of Hypothesis Testing The null hypothesis: the mean is equal to 3.0: The alternative hypothesis: the mean does not equal to 3.0:

© 2007 Thomson/South-Western. All rights reserved.21–12 An Example of Hypothesis Testing

© 2007 Thomson/South-Western. All rights reserved.21–13 EXHIBIT 21.3 A Hypothesis Test Using the Sampling Distribution of X under the Hypothesis µ = 3.0 — Critical Values Values that lie exactly on the boundary of the region of rejection.

© 2007 Thomson/South-Western. All rights reserved.21–14 Errors in Rejecting the Null Hypothesis Type I ErrorType I Error  An error caused by rejecting the null hypothesis when it is true; has a probability of alpha.  Practically, a Type I error occurs when the researcher concludes that a relationship or difference exists in the population when in reality it does not exist.  “There really are no monsters under the bed”

© 2007 Thomson/South-Western. All rights reserved.21–15 Errors in not Rejecting the Null Hypothesis Type II ErrorType II Error  An error caused by failing to reject the null hypothesis when the alternative hypothesis is true; has a probability of beta.  Practically, a Type II error occurs when a researcher concludes that no relationship or difference exists when in fact one does exist.  “There really are monsters under the bed”

© 2007 Thomson/South-Western. All rights reserved.21–16 EXHIBIT 21.4 Type I and Type II Errors in Hypothesis Testing

© 2007 Thomson/South-Western. All rights reserved.21–17 Choosing the Appropriate Statistical Technique Choosing the correct statistical technique requires considering:Choosing the correct statistical technique requires considering:  The type of question to be answered  The number of variables involved  The level of scale measurement Choice of statistical technique influences:Choice of statistical technique influences:  The research design  The type of data collected

© 2007 Thomson/South-Western. All rights reserved.21–18 Parametric versus Nonparametric Hypothesis Tests Parametric StatisticsParametric Statistics  Involve numbers with known, continuous distributions; when the data are interval or ratio scaled and the sample size is large, parametric statistical procedures are appropriate. Nonparametric StatisticsNonparametric Statistics  Appropriate when the variables being analyzed do not conform to any known or continuous distribution.

© 2007 Thomson/South-Western. All rights reserved.21–19 EXHIBIT 21.5 Univariate Statistical Choice Made Easy

© 2007 Thomson/South-Western. All rights reserved.21–20 The t-Distribution t-testt-test  A hypothesis test that uses the t-distribution.  A univariate t-test is appropriate when the variable being analyzed is interval or ratio. Degrees of freedom (d.f.)Degrees of freedom (d.f.)  The number of observations minus the number of constraints or assumptions needed to calculate a statistical term.

© 2007 Thomson/South-Western. All rights reserved.21–21 EXHIBIT 21.6 The t-Distribution for Various Degrees of Freedom

© 2007 Thomson/South-Western. All rights reserved.21–22 Calculating a Confidence Interval Estimate Using the t-Distribution

© 2007 Thomson/South-Western. All rights reserved.21–23 Calculating a Confidence Interval Estimate Using the t-Distribution (cont’d)

© 2007 Thomson/South-Western. All rights reserved.21–24 One-Tailed Univariate t-Tests One-tailed TestOne-tailed Test  Is appropriate when a research hypothesis implies that an observed mean can only be greater than or less than a hypothesized value.  Only one of the “tails” of the bell-shaped normal curve is relevant.  A one-tailed test can be determined from a two-tailed test result by taking half of the observed p-value.  When the researcher has any doubt about whether a one- or two-tailed test is appropriate, he or she should opt for the less conservative two-tailed test.

© 2007 Thomson/South-Western. All rights reserved.21–25 Two-Tailed Univariate t-Tests Two-tailed TestTwo-tailed Test  Tests for differences from the population mean that are either greater or less.  Extreme values of the normal curve (or tails) on both the right and the left are considered.  When a research question does not specify whether a difference should be greater than or less than, a two-tailed test is most appropriate.  When the researcher has any doubt about whether a one- or two-tailed test is appropriate, he or she should opt for the less conservative two-tailed test.

© 2007 Thomson/South-Western. All rights reserved.21–26 Univariate Hypothesis Test Utilizing the t-Distribution Example:Example:  Suppose that a production manager believes the average number of defective assemblies each day to be 20.  The factory records the number of defective assemblies for each of the 25 days it was opened in a given month.  The mean for defective assemblies was calculated to be 22, and the standard deviation to be 5.

© 2007 Thomson/South-Western. All rights reserved.21–27 Univariate Hypothesis Test Utilizing the t-Distribution: An Example The sample mean is equal to 20. The sample mean is equal not to 20.

© 2007 Thomson/South-Western. All rights reserved.21–28 Univariate Hypothesis Test Utilizing the t-Distribution: An Example (cont’d) The researcher desired a 95 percent confidence; the significance level becomes.05.The researcher desired a 95 percent confidence; the significance level becomes.05. The researcher must then find the upper and lower limits of the confidence interval to determine the region of rejection.The researcher must then find the upper and lower limits of the confidence interval to determine the region of rejection.  Thus, the value of t is needed.  For 24 degrees of freedom (n-1= 25-1), the t-value is

© 2007 Thomson/South-Western. All rights reserved.21–29 Univariate Hypothesis Test Utilizing the t-Distribution: An Example (cont’d) Lower limit = Upper limit =

© 2007 Thomson/South-Western. All rights reserved.21–30 Univariate Hypothesis Test Utilizing the t-Distribution: An Example (cont’d) Univariate Hypothesis Test t-Test

© 2007 Thomson/South-Western. All rights reserved.21–31 The Chi-Square Test for Goodness of Fit Chi-square ( χ 2 ) t-testChi-square ( χ 2 ) t-test  Tests for statistical significance  Is particularly appropriate for testing hypotheses about frequencies arranged in a frequency or contingency table. Goodness-of-Fit (GOF)Goodness-of-Fit (GOF)  A general term representing how well some computed table or matrix of values matches some population or predetermined table or matrix of the same size.

© 2007 Thomson/South-Western. All rights reserved.21–32 The Chi-Square Test for Goodness of Fit: An Example

© 2007 Thomson/South-Western. All rights reserved.21–33 The Chi-Square Test for Goodness of Fit: An Example (cont’d) χ² = chi-square statistics O i = observed frequency in the i th cell E i = expected frequency on the i th cell

© 2007 Thomson/South-Western. All rights reserved.21–34 Chi-Square Test: Estimation for Expected Number for Each Cell R i = total observed frequency in the i th row C j = total observed frequency in the j th column n = sample size

© 2007 Thomson/South-Western. All rights reserved.21–35 Hypothesis Test of a Proportion Hypothesis Test of a ProportionHypothesis Test of a Proportion  Is conceptually similar to the one used when the mean is the characteristic of interest but that differs in the mathematical formulation of the standard error of the proportion. π is the population proportion p is the sample proportion π is estimated with p

© 2007 Thomson/South-Western. All rights reserved.21–36 Hypothesis Test of a Proportion (cont’d)

© 2007 Thomson/South-Western. All rights reserved.21–37 Key Terms and Concepts Univariate statistical testUnivariate statistical test bivariate statistical testbivariate statistical test multivariate statistical testmultivariate statistical test significance levelsignificance level critical valuescritical values p-valuep-value censuscensus type I errortype I error type II errortype II error parametric statisticsparametric statistics nonparametric statisticsnonparametric statistics t-distributiont-distribution degrees of freedomdegrees of freedom t-testt-test chi-square ( χ 2 ) testchi-square ( χ 2 ) test goodness of fitgoodness of fit hypothesis test of a proportionhypothesis test of a proportion

© 2007 Thomson/South-Western. All rights reserved.21–38 CASE EXHIBIT 21.1–1 Miles per Gallon Information