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Chapter 17
Inferential Statistics
Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins
Question
Tell whether the following statement is true or false:
Inferential statistics are based on laws of probability.
Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins
Answer
True
Inferential statistics, which are based on laws of
probability, allow researchers to make inferences about a
population based on data from a sample; they offer a
framework for deciding whether the sampling error that
results from sampling fluctuations is too high to provide
reliable population estimates.
Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins
Inferential Statistics
• A means of drawing conclusions about
a population given data from a sample
• Based on laws of probability
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Sampling Distribution of the Mean
• A theoretical distribution of means for an infinite
number of samples drawn from the same
population
• Is always normally distributed
• Has a mean that equals the population mean
• Has a standard deviation (SD) called the standard
error of the mean (SEM)
• SEM is estimated from a sample SD and the
sample size
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Sampling Distribution
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Question
Tell whether the following statement is true or false:
Point estimation through statistical procedures enables
researchers to make objective decisions about the
validity of their hypotheses.
Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins
Answer
False
Hypothesis testing through statistical procedures enables
researchers to make objective decisions about the
validity of their hypotheses. Point estimation provides a
single descriptive value of the population estimate
Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins
Statistical Inference—Two Forms
• Estimation of parameters
• Hypothesis testing (more common)
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Estimation of Parameters
• Used to estimate a single parameter
• Two forms of estimation:
– Point estimation
– Interval estimation
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Point Estimation
Calculating a single statistic to estimate
the population
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Interval Estimation
• Calculating a range of values within
which the parameter has a specified
probability of lying:
– A confidence interval (CI) is
constructed around the point
estimate
– The upper and lower limits are
confidence limits
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Hypothesis Testing
• Based on rules of negative inference:
research hypotheses are supported if null
hypotheses can be rejected
• Involves statistical decision making to
either:
– Accept the null hypothesis, or
– Reject the null hypothesis
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Hypothesis Testing (cont’d)
• Researchers compute a test statistic with their data,
then determine whether the statistic falls beyond
the critical region in the relevant theoretical
distribution
• If the value of the test statistic indicates that the
null hypothesis is “improbable,” the result is
statistically significant
• A nonsignificant result means that any observed
difference or relationship could have resulted from
chance fluctuations
Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins
Question
Tell whether the following statement is true or false:
Type II error occurs when a null hypothesis is incorrectly
rejected (a false positive).
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Answer
False
A Type I error occurs when a null hypothesis is incorrectly
rejected (a false positive). A Type II error occurs when a
null hypothesis is wrongly accepted (a false negative).
Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins
Statistical Decisions Are Either Correct or
Incorrect
Two types of incorrect decisions:
• Type I error: a null hypothesis is rejected when
it should not be rejected
– Risk of a Type I error is controlled by the
level of significance (alpha), that is,  = .05
or .01.
• Type II error: failure to reject a null hypothesis
when it should be rejected
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Hypotheses Testing
• Test statistic
• Critical region
• Statistically significant
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One-Tailed and Two-Tailed Tests
Two-tailed tests
Hypothesis testing in which both ends of the
sampling distribution are used to define the region
of improbable values
One-tailed tests
Critical region of improbable values is entirely in one
tail of the distribution—the tail corresponding to the
direction of the hypothesis
Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins
Parametric Statistics
• Involve the estimation of a parameter
• Require measurements on at least an
interval scale
• Involve several assumptions (e.g., that
variables are normally distributed in
the population)
Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins
Nonparametric Statistics (DistributionFree Statistics)
• Do not estimate parameters
• Involve variables measured on a
nominal or ordinal scale
• Have less restrictive assumptions about
the shape of the variables’ distribution
than parametric tests
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Overview of Hypothesis-Testing
Procedures
• Select an appropriate test statistic
• Establish the level of significance (e.g.,  =
.05)
• Select a one-tailed or a two-tailed test
• Compute test statistic with actual data
• Calculate degrees of freedom (df) for the
test statistic
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Overview of Hypothesis-Testing
Procedures (cont’d)
• Obtain a tabled value for the statistical
test
• Compare the test statistic to the tabled
value
• Make decision to accept or reject null
hypothesis
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Groups
• Independent groups compare separate
groups of people
• Dependent groups compare the same group
of people over time or conditions
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Commonly Used Bivariate Statistical Tests
1. t-Test
2. Analysis of variance (ANOVA)
3. Pearson’s r
4. Chi-square test
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t-Test
Tests the difference between two means
– t-Test for independent groups
(between subjects)
– t-Test for dependent groups (within
subjects)
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Analysis of Variance (ANOVA)
• Tests the difference between 3+
means
– One-way ANOVA
– Multifactor (e.g., two-way) ANOVA
– Repeated measures ANOVA (within
subjects)
Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins
Correlation
• Pearson’s r, a parametric test
• Tests that the relationship between two
variables is not zero
• Used when measures are on an interval
or ratio scale
Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins
Chi-Square Test
• Tests the difference in proportions in
categories within a contingency table
• A nonparametric test
Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins
Power Analysis
• A method of reducing the risk of Type II
errors and estimating their occurrence
• With power = .80, the risk of a Type II
error () is 20%
• Method is frequently used to estimate how
large a sample is needed to reliably test
hypotheses
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Power Analysis (cont’d)
Four components in a power analysis:
1. Significance criterion (α)
2. Sample size (N)
3. Population effect size—the magnitude of the
relationship between research variables (γ)
4. Power—the probability of obtaining a
significant result (1-β)
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