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Chapter 13:

Introductory Inferential Statistics I:

Hypothesis Testing with Two Means



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

probability: the tendency or likelihood with which an event occurs in a population

    B. Using Probability Distributions

probability distributions: distributions that represent the theoretical patterns of expected “values of a random variable and of the probabilities of occurrence of these values” (Upton & Cook, 2002, p. 292)

II. Reasoning With Statistical Hypothesis Testing

research hypothesis: (symbolized H1) “an expectation about events based on generalizations of the assumed relationship between variables” (Tuckman, 1999, p. 74)

   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)    

null hypothesis: (symbolized Ho) a statistical hypothesis that states that there is no relationship between variables

    B.   Decisions in Testing Statistical Hypotheses

--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.   





--determinism:  the notion that the general course of events is determined by structures deemed to be fundamental

              1.  Finding Unusual Occurrences


critical region: a portion of a probability distribution which, if our test statistic falls in that zone, will cause rejection of the null hypothesis

statistically significant difference or relationship: as a result of a test of statistical significance, finding a relationship or effect size that is unlikely to have been found from random sampling error if the null hypothesis were true.

critical value: in statistical significance testing, the line that divides the critical region from the rest of the distribution
standard error of the mean: the standard deviation of a distribution of means

standard error: the standard deviation of a distribution of elements other than raw scores

             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 the Null




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

            2.  Computing the Test Statistics

test statistic: in statistical hypothesis testing, a number computed from a statistical formula, “which is a function of the observations in a random sample” (Upton & Cook, 2002, p. 165)

            3.  Finding the Critical Value

one-tailed tests: a test using a one-tailed or directional material hypothesis that states the form of predicted differences and requires using a critical range on one side of a probability distribution.

two-tailed test: a test of a two-tailed or nondirectional material hypothesis that does not state the form of predicted differences and requires using a critical range on both sides of a probability distribution.

         4.  Rejecting or Failing to Reject the Null Hypothesis


III.  Comparisons of Two Means: The t test

parametric testing: statistical significance 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
   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 participants in some
          conditions more than others).

    A.  Forms of the t Test
          1.  One-Sample t Test

one-sample t test: an application of the t test that examines when a new sample mean differs from a known population mean, under conditions where the population standard deviation is unavailable

          2.   t Test for Independent Samples

t test for independent samples: an application of the t test that compares the means or two sample groups (one of which often is a control or current condition)
pooled standard deviation (sp): the square root of the average of variances in subgroups involved in comparisons

          3.   t Test for Dependent Samples

t test for dependent samples: an application of the t test that compares the mean difference in two scores in which participants are matched or sampled twice

          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 statistical significance of a correlation and zero

     B. Determining Effect Sizes