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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
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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.
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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.
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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.
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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.*
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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.
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EXHIBIT 21.1p-Values and Statistical Tests
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EXHIBIT 21.2
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.
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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:
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An Example of Hypothesis Testing
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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.
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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.
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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.
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EXHIBIT 21.4Type I and Type II Errors in Hypothesis Testing
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Choosing the Appropriate Statistical TechniqueChoosing the
correct statistical technique requires considering:Type of question
to be answeredNumber of variables involvedLevel of scale
measurement
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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.
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EXHIBIT 21.5Univariate Statistical Choice Made Easy
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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.
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EXHIBIT 21.6The t-Distribution for Various Degrees of
Freedom
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Calculating a Confidence Interval Estimate Using the
t-Distribution
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Calculating a Confidence Interval Estimate Using the
t-Distribution (contd)
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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.
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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.
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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.*
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Univariate Hypothesis Test Utilizing the t-Distribution: An
Example*The sample mean is equal to 20.The sample mean is equal not
to 20.
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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.*
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Univariate Hypothesis Test Utilizing the t-Distribution: An
Example (contd)
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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.
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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.
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The Chi-Square Test for Goodness of Fit: An Example
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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
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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
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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