Discussion Question
yury12082Chapter 18 Inferential Statistics
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Question #1
Tell whether the following statement is true or false:
Inferential statistics are based on laws of probability.
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Answer to Question #1
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.
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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 #2
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.
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Answer to Question #2
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.
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Estimation of Parameters
Statistical inference—Two forms
Estimation of parameters
Hypothesis testing (more common)
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Confidence Intervals
Used to estimate a single parameter
Two forms of estimation
Point estimation
Calculating a single statistic to estimate the population
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 #1
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 #2
Researchers compute a test statistic with their data and 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.
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Question #3
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 to Question #3
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).
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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.
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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)
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Nonparametric Statistics (Distribution-Free 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 #1
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 #2
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
t-test
Analysis of variance (ANOVA)
Pearson’s r
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 three or more means
One-way ANOVA
Multifactor (e.g., two-way) ANOVA
Repeated-measures ANOVA (within subjects)
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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
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Chi-Square Test
Tests the difference in proportions in categories within a contingency table
A nonparametric test
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Power Analysis #1
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 #2
Four components in a power analysis
Significance criterion ()
Sample size (N)
Population effect size—the magnitude of the relationship between research variables (γ)
Power—the probability of obtaining a significant result (1 − β)
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