Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
CHAPTERS 16-17 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. Steps Involved in Hypothesis Testing Formulate H0 and H1 Select Appropriate Test Choose Level of Significance, Collect Data and Calculate Test Statistic Determine Probability Associated with Test Statistic Determine Critical Value of Test Statistic TSCR Compare with Level of Significance, Determine if TSCR falls into (Non) Rejection Region Reject, or Fail to Reject H0 Draw Marketing Research Conclusion Other Issues in Hypothesis Testing Types of errors in hypothesis testing. Type I error Type II error Rejection of a null hypothesis when, in fact, it is true Fail to reject the null hypothesis when, in fact, it is false Other Issues in Hypothesis Testing (cont.) Type I and Type II Errors Actual State of the Null Hypothesis Ho is true Ho is false Fail to Reject Ho Reject Ho Correct (1 - ) no error Type II error (b) Type I Error () Correct (1 - b) no error Probabilities of Type I & Type II Error 95% of Total Area = 0.05 = 15 Z = 1.645 Critical Value 99% of of Z Total Area Z b = 0.01 = 17 Z b = -2.33 Z 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= population mean under the ( ) (sample mean) null hypothesis estimated standard error of the mean Hypotheses About One Mean(cont.) T-test: Hypothesis test about a single mean if the sample is too small (n < 30) to use the Ztest. Calculation of the Test Statistic: t= population mean specified (sample mean) - under the null hypothesis estimated standard error of the mean ( ) Probability of z with a One-Tailed Test Shaded Area = 0.9664 Unshaped Area = 0.0336 0 z = 1.83 A Broad Classification of Hypothesis Tests Hypothesis Tests Tests of Differences Tests of Association Distributions Means Proportions Median/ Rankings 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. Gender and Internet Usage Sex Internet Usage Male Female Row Total Light (1) 5 10 15 Heavy (2) 10 5 15 15 15 Column Total Internet Usage by Sex Sex Internet Usage Male Female Light 33.3% 66.7% Heavy 66.7% 33.3% Column total 100% 100% Sex by Internet Usage Internet Usage Sex Light Heavy Total Male 33.3% 66.7% 100.0% Female 66.7% 33.3% 100.0% Purchase of Fashion Clothing by Marital Status Purchase of Fashion Clothing Current Marital Status Married Unmarried High 31% 52% Low 69% 48% Column 100% 100% 700 300 Number of respondents Purchase of Fashion Clothing by Marital Status and Gender Pur chase of Fashion Clothing Sex Male Marr ied Female High 35% Unmarried Not Mar r ied 40% Mar r ied Low 65% 60% 75% 40% Column totals Number of cases 100% 100% 100% 100% 400 120 300 180 25% Unmarried Not Mar r ied 60% Ownership of Expensive Automobiles by Education Level Own Expensive Automobile Education College Degr ee No College Degr ee Yes 32% 21% No 68% 79% Column totals 100% 100% 250 750 Number of cases Desire to Travel Abroad by Age Desir e to Tr avel Abr oad Age Less than 45 45 or Mor e Yes 50% 50% No 50% 50% Column totals 100% 100% 500 500 Number of respondents Eating Frequently in Fast Food Restaurants by Family Size Eat Fr equently in Fast Food Restaurants Family Size Small Lar ge Yes 65% 65% No 35% 35% Column totals 100% 100% 500 500 Number of cases Ownership of Expensive Automobiles by Education Level and Income Levels Own Expensive Automobile Low Income College Degr ee Income High Income Yes 20% No College Degr ee 20% College Degr ee 40% No College Degr ee 40% No 80% 80% 60% 60% Column totals Number of r espondents 100% 100% 100% 100% 100 700 150 50 Eating Frequently in Fast Food Restaurants by Family Size & Income Eat Fr equently in Fast Food Restaurants Low Family size Income High Family size Small Large Small Lar ge Yes 65% 65% 65% 65% No 35% 35% 35% 35% Column totals Number of Respondents 100% 100% 100% 100% 250 250 250 250 Desire to Travel Abroad by Age and Gender Desir e to Tr avel Abr oad Sex Male Age Female Age < 45 >=45 <45 >=45 Yes 60% 40% 35% 65% No 40% 60% 65% 35% Column totals Number of Cases 100% 100% 100% 100% 300 300 200 200 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 Fail to Reject H0 Reject H0 Critical Value, or table value of X2 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. A Classification of Hypothesis Testing Procedures for Examining Differences Hypothesis Tests Non-parametric Tests (Nonmetric Tests) Parametric Tests (Metric Tests) One Sample * t test * Z test Two or More Samples Independent Samples * Two-Group t test * Z test Paired Samples * Paired t test One Sample * Chi-Square * K-S * Runs * Binomial Two or More Samples Independent Samples * Chi-Square * Mann-Whitney * Median * K-S Paired Samples * Sign * Wilcoxon * McNemar * Chi-Square