CHAPTERS HYPOTHESIS TESTING, AND DETERMINING AND INTERPRETING BETWEEN TWO VARIABLES

Important Topics of These Chapters Hypothesis and testing. Steps involved in hypotheses testing. Type I and Type II errors. Independent and related samples. Degrees of freedom. Hypotheses about single mean. Cross-tabulations. Goodness of fit and chi-square tests. How to interpret a chi-square result.

Hypotheses and Hypothesis Testing Hypotheses: Assumptions, intuition, prior knowledge or theories that a researcher or manager makes statements about population parameter under study. Most commonly takes the form of exact specification as what the population parameter value is. Hypothesis Testing: Statistical procedure used to ‘accept’ or ‘reject’ the hypothesis based on sample information. For hypothesis testing, sample is the most current information about population.

Hypothesis Testing (cont.) Statement of hypotheses: Null Hypothesis(Ho:): Stated hypothesis: Mean Income of population is equal to $36,000. Alternative Hypothesis (Ha:): Alternative that tested against the ‘Null hypothesis’. Mean income of population is not equal to $36,000. (Two tails test).,or Directional Hypothesis: Mean income of population is less than $36,000.(one tail test), or Mean income of population is higher than $36,000. (one tail test).

Hypothesis testing (cont.) Statistical technique to test the Null and Alternative Hypotheses: Cross tabulation, or other available statistical techniques. Decision rule as the basis for determining whether to reject or fail to reject the null hypothesis: Computed and table value of test statistic (for cross-tabulation, it is the table value of Chi-Square statistics at certain degree of freedom (d.f.= c-1 X r-1), against to computed value of Chi-Square statistic. Significance level: At.01 level (99% confidence), or at.05 level (95% confidence), or at.10 level (90% confidence). Reject, or fail to reject the Null Hypothesis by basing upon decision rule. State the conclusion from the perspective of the original research problem or question.

Formulate H 0 and H 1 Steps Involved in Hypothesis Testing Select Appropriate Test Collect Data and Calculate Test Statistic Determine Probability Associated with Test Statistic Choose Level of Significance,  Draw Marketing Research Conclusion Reject, or Fail to Reject H 0 Determine Critical Value of Test Statistic TSCR Determine if TSCR falls into (Non) Rejection Region Compare with Level of Significance, 

Other Issues in Hypothesis Testing Types of errors in hypothesis testing. Type II error Fail to reject the null hypothesis when, in fact, it is false Type I error Rejection of a null hypothesis when, in fact, it is true

Other Issues in Hypothesis Testing (cont.) Type I and Type II Errors Actual State of the Null Hypothesis Fail to Reject HoReject Ho Ho is trueCorrect (1 -  ) no errorType I Error (  ) Ho is falseType II error (  )Correct (1 -  ) no error

Probabilities of Type I & Type II Error 99% of Total Area Critical Value of Z   = 15   = 17  = 0.01 = Z  = Z  Z Z 95% of Total Area  = 0.05

Other Issues in Hypothesis Testing (cont.) Accepting Ho or Failing to Reject (FTR) Ho: Researchers often fail to make a distinction between accepting and failing to reject (FTR) Ho. One-Tailed Test or Two-Tailed Test: The decision of whether to use a one-tailed test or a two-tailed test depends on the nature of the situation and what you are trying to demonstrate when you stated the Null Hypothesis.

Hypothesis Tests Independent versus Related Samples Independent Samples: Samples in which measurement of a variable in one population has no effect on the measurement of the variable in another. Related Samples: Samples in which the measurement of a variable in one population may influence the measurement of the variable in another.

Hypothesis Tests (cont.) Degrees of Freedom: Degrees of freedom are the number of observations in a statistical problem that are not restricted or are free to vary. The number of degrees of freedom (d.f.) is equal to the number of observations minus the number of assumptions or constraints necessary to calculate a statistic.

Hypotheses About One Mean Z-test: Hypothesis test about a single mean if the sample is large enough (n > 30) and drawn from a normal population. Calculation of the Test Statistic: Z = estimated standard error of the mean (sample mean) - population mean under the null hypothesis ()

Hypotheses About One Mean(cont.) T-test: Hypothesis test about a single mean if the sample is too small (n < 30) to use the Z- test. Calculation of the Test Statistic: t = estimated standard error of the mean (sample mean) - population mean specified under the null hypothesis ()

Unshaped Area = Probability of z with a One-Tailed Test Shaded Area = z =

Hypothesis Tests Distributions A Broad Classification of Hypothesis Tests Tests of Association Tests of Differences Median/ Rankings Means Proportions

Cross-Tabulations Monotonic relationships: Researcher can assign only a general direction (increase or decrease) between two variables. Non-monotonic relationships: The presence(or absence) of one variable is systematically associated with the presence(or absence) of another variable. Cross tabulation and associated chi-square: It used to assess whether a non-monotonic relationship exits between two nominal-scaled variables.

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Goodness of Fit Chi-Square Test: Test of the goodness of fit between the observed distribution and the expected distribution of a variable. Statement of Hypotheses: Ho: There is not an association (relationship) between variable ‘X’ and variable ‘Y’. Ha: There is an association (relationship) between variable ‘X’ and variable ‘Y’.

Chi-Square Analysis The computed Chi-Square value: n (observed - Expected) X2 = i - 1 Expected Where: Observed: Observed frequency of cell i. Expected: Expected frequency of cell I. n: number of cells.

Chi-Square Analysis (cont.) The Chi-Square Distribution: Table value or critical value of Chi-Square at certain degree of freedom (d.f). Degrees of freedom (d.f.) for Chi-Square statistics: (r-1) (c-1)

Chi-Square Distribution: One Tail Test Reject H 0 Fail to Reject H 0 Critical Value, or table value of X2 22

How to Interpret a Chi-Square Result It yields the probability that researcher find evidence in support of the null hypothesis. It should be pointed out that whether or not a non-monotonic relationship exits between variable ‘X’ and variable “Y’. The chi-square test does not indicate the nature or direction of association between the two variables. The chi-square test indicated the strength of association that exits between two variables.

Independent Samples One Sample Two or More Samples One Sample Two or More Samples Paired Samples Independent Samples Paired Samples * t test * Z test * Chi-Square * K-S * Runs * Binomial * Two-Group t test * Z test * Paired t test * Chi-Square * Mann-Whitney * Median * K-S * Sign * Wilcoxon * McNemar * Chi-Square Hypothesis Tests Parametric Tests (Metric Tests) Non-parametric Tests (Nonmetric Tests) A Classification of Hypothesis Testing Procedures for Examining Differences