Chapter 21 Univariate Statistical Analysis © 2010 South-Western/Cengage Learning. All rights...
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Transcript of Chapter 21 Univariate Statistical Analysis © 2010 South-Western/Cengage Learning. All rights...
LEARNING OUTCOMESImplement the hypothesis-testing procedureUse p-values to assess statistical significanceTest a hypothesis about an observed mean compared to some standardKnow the difference between Type I and Type II errorsKnow when a univariate 2 test is appropriate and how to conduct oneAfter studying this chapter, you should be able to
Hypothesis TestingTypes of HypothesesRelational hypothesesExamine how changes in one variable vary with changes in another.Hypotheses about differences between groupsExamine how some variable varies from one group to another.Hypotheses about differences from some standardExamine how some variable differs from some preconceived standard. These tests typify univariate statistical tests.
Types of Statistical AnalysisUnivariate Statistical AnalysisTests of hypotheses involving only one variable.Testing of statistical significanceBivariate Statistical AnalysisTests of hypotheses involving two variables.Multivariate Statistical AnalysisStatistical analysis involving three or more variables or sets of variables.
The Hypothesis-Testing ProcedureProcessThe 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.
Statistical Analysis: Key TermsHypothesisUnproven proposition: a supposition that tentatively explains certain facts or phenomena.An assumption about nature of the world.Null HypothesisStatement about the status quo.No difference in sample and population.Alternative HypothesisStatement that indicates the opposite of the null hypothesis.*
Significance Levels and p-valuesSignificance LevelA 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-valueProbability 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.
EXHIBIT 21.1p-Values and Statistical Tests
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.
An Example of Hypothesis TestingThe null hypothesis: the mean is equal to 3.0:The alternative hypothesis: the mean does not equal to 3.0:
An Example of Hypothesis Testing
EXHIBIT 21.3A Hypothesis Test Using the Sampling Distribution of X under the Hypothesis = 3.0Critical Values Values that lie exactly on the boundary of the region of rejection.
Type I and Type II ErrorsType I ErrorAn 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.
Type I and Type II Errors (contd)Type II ErrorAn 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.
EXHIBIT 21.4Type I and Type II Errors in Hypothesis Testing
Choosing the Appropriate Statistical TechniqueChoosing the correct statistical technique requires considering:Type of question to be answeredNumber of variables involvedLevel of scale measurement
Parametric versus Nonparametric TestsParametric StatisticsInvolve numbers with known, continuous distributions.Appropriate when:Data are interval or ratio scaled.Sample size is large.Nonparametric StatisticsAppropriate when the variables being analyzed do not conform to any known or continuous distribution.
EXHIBIT 21.5Univariate Statistical Choice Made Easy
The t-Distributiont-testA 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.)The number of observations minus the number of constraints or assumptions needed to calculate a statistical term.
EXHIBIT 21.6The t-Distribution for Various Degrees of Freedom
Calculating a Confidence Interval Estimate Using the t-Distribution
Calculating a Confidence Interval Estimate Using the t-Distribution (contd)
One-Tailed Univariate t-TestsOne-tailed TestAppropriate 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 there is any doubt about whether a one- or two-tailed test is appropriate, opt for the less conservative two-tailed test.
Two-Tailed Univariate t-TestsTwo-tailed TestTests 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.
Univariate Hypothesis Test Utilizing the t-DistributionExample:Suppose a Pizza Inn manager believes the average number of returned pizzas each day to be 20. The store records the number of defective assemblies for each of the 25 days it was opened in a given month.The mean was calculated to be 22, and the standard deviation to be 5.*
Univariate Hypothesis Test Utilizing the t-Distribution: An Example*The sample mean is equal to 20.The sample mean is equal not to 20.
Univariate Hypothesis Test Utilizing the t-Distribution: An Example (contd)The researcher desired a 95 percent confidence; the significance level becomes 0.05.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 2.064.*
Univariate Hypothesis Test Utilizing the t-Distribution: An Example (contd)
Univariate Hypothesis Test Utilizing the t-Distribution: An Example (contd)Univariate Hypothesis Test t-TestThis is less than the critical t-value of 2.064 at the 0.05 level with 24 degrees of freedom hypothesis is not supported.
The Chi-Square Test for Goodness of FitChi-square (2) testTests for statistical significance.Is particularly appropriate for testing hypotheses about frequencies arranged in a frequency or contingency table.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.
The Chi-Square Test for Goodness of Fit: An Example
The Chi-Square Test for Goodness of Fit: An Example (contd) = chi-square statisticsOi = observed frequency in the ith cellEi = expected frequency on the ith cell
Chi-Square Test: Estimation for Expected Number for Each CellRi = total observed frequency in the ith rowCj = total observed frequency in the jth columnn = sample size
Hypothesis Test of a ProportionHypothesis Test of a ProportionIs 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. is the population proportionp is the sample proportion is estimated with p