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Chapter 13:

Introductory Inferential Statistics I:

Hypothesis Testing with Two Means

 

Outline

 

Concepts

 

I.  Using Probability
    Distributions to Play
    the Odds
    A. Using the Statistics
         of Probability and
         Inference
 

probability: the frequency that an event occurs in a population

    B. Using Probability
         Distributions
probability distributions: distributions that represent the theoretical patterns of expected sample data
II. Reasoning With
   Statistical Hypothesis
   Testing
research hypothesis: (symbolized H1) an expectation about events based on generalizations of the assumed relationship between variables
   A. Determining
        Statistical
        Hypotheses
        --if the null
           hypothesis is
           rejected, the
           research
           hypothesis
           may be tenable
           and the
           researchers
           conclude that a
           relationship exists
           among variables;
        --if the null
           hypothesis is not
           rejected (logically
           we cannot claim
           to "accept" a null
           hypothesis], then
           researchers
           make no claim
           that a
           relationship
           exists among
           variables).
        --For the sake of
          argument, we
          assume that there
          are no relationships
          among variables
          (temporarily
          assuming the null
          hypothesis is true).
          Then, we look at
          our data and ask:
          "how likely is it that
          we could find
          results such as we
          observed in our
          samples if no
          relationships
          existed in the
          population?"
       --If finding results
         such as ours is quite
         probable when
         sampling from a
         population in which
         no relationships
         existed, we agree
         to continue
         assuming that any
         differences are just
         random;
      --if it is very
        improbable that
        results such as ours
        could be found by
        sampling from a
        population in which
        no relationships
        exist, we reject the
        assumption that any
        differences are just
        random.           
null hypothesis: (symbolized Ho) states that there is no relationship between variables
    B.   Decisions in
          Testing
          Statistical
          Hypotheses
         1.  Finding
              Unusual
              Occurrences
 

 

critical region: a portion of a probability distribution which, if our test statistic falls in that zone, will cause researchers to claim a "significant difference" ("significant difference or relationship" is one that is beyond what might be expected to occur by chance alone)
critical value: the line that divides the critical region from the rest of the distribution
central limit theorem: a statement that a sampling distribution of means tends toward normal distribution with increased sample size regardless of the shape of the parent population
standard error of the mean: the standard deviation of a distribution of means (whenever a standard deviation is based on measures other than raw scores, it is called the "standard error" of that measure)
determinism: the notion that the general course of events is determined by structures deemed to be fundamental

         2. Choice and
              Errors in
              Testing
              Statistical
              Hypotheses
Type I error: incorrectly rejecting the null hypothesis (claiming that the null hypothesis is untrue when it is true) alpha risk: the probability of committing a Type I error (for alpha risk of .05, 5% of the distribution is in the critical region; for alpha risk of .01, only 1% of the distribution is established as the critical region, and so forth)
Type II error: failing to detect a relationship that is present (failing to reject the null hypothesis when the null hypothesis is false)
beta risk: the probability of committing a Type II error
power: the probability of rejecting the null hypothesis correctly
    C. The Process of
         Examining
         Statistical
         Hypotheses
         1. Determine
              Decision
              Rule for
              Rejecting
              Null
              Hypothesis
statistical significance: when differences are claimed following a statistical test; "statistical significance"  does not mean that an important relationship exists, but just that such results are not likely to be due to chance alone
         2. Computing
              the Test
              Statistic
test statistic: a number computed from a statistical formula to test the statistical hypothesis
         3.   Finding the
              Critical Value
one-tailed tests: a test using a distribution in which the critical region lies on only one side of a two-tailed distribution
two-tailed tests: a test using a distribution in which the critical region lies on both sides of a two-tailed distribution
         4.   Rejecting or
              Failing to
              Reject the
              Null
              Hypothesis
III.  Comparisons of Two
     Means: the t test
parametric testing: tests that make assumptions about populations from which the data were drawn;
--assumptions underlying
   parametric tests:
   1.  randomization;
   2.  measurement of the
        dependent variable
        on the interval or
        ratio level;
   3.  underlying normal
        distribution of events
        in a population;
   4.  homogeneous
        variances of the groups
        compared (this
        homogeneity
        requirement means
        that any differences in
        variances should be
        within the limits of
        sampling error).
        --if sample sizes of
          groups compared are
          equal, violating the
          assumptions of normal
          distributions and
          homogenous variances
          are relatively
          unimportant;
       --unequal variances can
          stem from:  ceiling or
          floor effects (which
          occur when scores are
          so high [or so low] in
          samples, that the data
          are "crunched up" [or
          down] near the end of
          the measurement
          range); subjects by
           treatments interaction
           (which indicates that
           at least one additional
           variable uncontrolled
           by the researcher has
           introduced systematic
           variation, influencing
           subjects in some
           conditions more than
           others).
    A.  Forms of the t Test
          1. One-Sample t Test
one-sample t test: used to examine if a (known population mean comparison) new sample mean differs from a known population mean, under conditions where the population standard deviation is unavailable
degrees of freedom: the number of events in a study minus the number of population parameters estimated from samples
          2.   t Test for Independent
               Samples
t test for independent samples: used to compare two sample groups to each other (one of which often is a control or current condition)
pooled standard deviation (sp): an average standard deviation from different sample groups
          3.   t Test for Dependent
               Samples
t test for dependent samples: used to compare the mean difference in two scores for each individual in the sample
          4.  t Test for the Difference
               between Zero and an
               Observed Correlation
t test for the difference between zero and an observed correlation: used to test the significance of a correlation
     B. Determining Effect Sizes