R. E. Wyllys Copyright 2003 by R. E. Wyllys Last revised 2003 Jan 15 LIS 397.1 Introduction to Research in Library and Information Science Statistical Hypotheses R. E. Wyllys Copyright 2003 by R. E. Wyllys Last revised 2003 Jan 15 School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science

Hypotheses Hypotheses state a relationship among two or more variables Hypotheses may be stated in positive or negative terms Hypotheses must be capable of being tested as to whether they are “true” or “false” School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science

Two Types of Hypotheses General Hypotheses Concern variables directly related to the problem being studied Statistical Hypotheses Are a subclass of general hypotheses Are tools Are used in efforts to determine whether general hypotheses are true or false School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science

Statistical Hypothesis Makes a to-be-tested statement about either The kind of probability distribution that a certain variable obeys; or The value of a population parameter (average, total, proportion, etc.) Must be one of a relatively small number of standard types of statistical hypothesis School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science

Statistical Hypothesis vs. State of Nature DECISION H is true H is false Accept H OK error Reject H error OK School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science

Null Hypothesis vs. State of Nature Are the erroneous decisions of equal importance? If not (the usual case), then arrange the wording of the hypothesis so that the more serious error occurs when the hypothesis is true but you decide that it is false. I.e., arrange things so that the more serious error occurs when you reject a true hypothesis. This arrangement yields the Null Hypothesis, H0. STATE OF NATURE DECISION H0 is true H0 is false Accept H0 OK Type II error Reject H0 Type I error OK School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science

Null Hypothesis & Probabilities P(Type I error) =  = “level of significance of the test” = “risk of the test” = “alpha of the test” P(Type II error) =  P(not making Type II error) = 1 -  = “power of the test” = P(correctly recognizing hypothesis as false when it is false) School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science

Null Hypothesis & Probabilities Primary objective: Avoid more serious error; i.e., ensure that  is small Secondary objective: Increase chance of recognizing hypothesis as false if it is false; i.e., increase power of test, 1- For a given , the only way to increase power is to increase size of sample Power is very hard to determine; in practice,  gets almost all the attention School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science

Test of a Statistical Hypothesis Statements of Null hypothesis, H0 Alternative hypothesis (often simply negation of H0) Level of significance,  “Critical region”, i.e., what outcomes will lead to rejection of H0 (in practice, this usually means stating threshold value from appropriate statistical table) School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science

Common Types of Single-Variable Statistical Hypotheses Population mean is some number: “Average daily circulation total is 123” H0: 1 = 2 Means of two populations are equal: “Average cost per online search using Service A = average cost using Service B” School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science

Common Types of Single-Variable Statistical Hypotheses H0: 1 = 2 = 3 = ..., etc. Means of Populations 1, 2, 3, ..., etc. are all equal: “Average number of books borrowed per student per semester is the same for freshmen, sophomores, juniors, and seniors.” School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science

Common Types of Two-Variable Statistical Hypotheses H0: XY = 0 Variables X and Y are not correlated in the population: “There is no correlation between the age and the salary of a typical librarian” H0: Categorical variables X and Y are not associated: “There is no association between the sex of a library patron and the type of book the patron prefers” School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science

Standardized Tests of Statistical Hypotheses To each type of statistical hypothesis corresponds a particular standardized test procedure or procedures Each test procedure includes a formula, the “test statistic” You place, into the test statistic, data from observed sample or samples obtain a number, the observed value of the test statistic School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science

Standardized Tests of Statistical Hypotheses (cont'd) Traditional Method: Compare absolute value of observed value of test statistic against threshold value from pertinent table If |test statistic|  tabled threshold Accept H0 If |test statistic| > tabled threshold Reject H0 Computer-Era Method: Use probability of getting observed value of test statistic when the null hypothesis H0 is true (OVTSWNHT) If P(OVTSWNHT)   Accept H0 If P(OVTSWNHT) <  Reject H0 School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science

Comparing the Traditional and Computer-Era Methods Part of Excel’’s output for “A Worked Example” from pp. 88-90 of Hinton School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science

Evidence in the Sample is Weighed against Risk in order to Tip the Balance toward Acceptance or Rejection School of Information - The University of Texas at Austin LIS 397.1, Introduction to Research in Library and Information Science