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 H_{1}) 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 H_{o}) 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) 
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 
onetailed tests: a test using a distribution in which
the critical region lies on only one side of a twotailed distribution twotailed tests: a test using a distribution in which the critical region lies on both sides of a twotailed 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. OneSample t Test 
onesample 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 (s_{p}): 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 