STA-3123: Statistics for Behavioral and Social Sciences II. Text Book: McClave and Sincich, 12 th edition. Contents and Objectives

Transcription

1 STA-3123: Statistics for Behavioral and Social Sciences II Text Book: McClave and Sincich, 12 th edition Contents and Objectives Initial Review and Chapters 8 14 (Revised: Aug. 2014)

2 Initial Review on STA-2122 A. Probability Models Use of the Standard Normal table (z-table) Use of the t-table B. Basic Concepts in Inferential Statistics Population Sample Representative sample Sampling techniques. Simple random sampling Statistical inference Parameters and Statistics. Sampling distribution Estimation Hypothesis testing

3 Chapter 8: Hypothesis Testing based on a single sample Elements of hypothesis testing Research hypothesis Statistical hypotheses: null and alternative Test statistic (TS) Rejection region (RR). Location and critical values Type I and II errors. Probabilities Alpha and Beta Significance level of the test p-values P-values Interpreting p-values Computing p-values with the z-table Bracketing p-values with the t-table Two approaches for testing hypotheses Traditional: TS vs. RR Alternate: p-value vs. significance level Conducting tests of hypotheses about a (1) Population mean using a large sample: z-test (2) Population mean using a small sample: t-test (3) Population proportion using a large sample: z-test Reading and interpreting statistical software (SPSS) output

4 Chapter 9: Inferences based on two samples - Test of Hypotheses about two population means involving independent samples Case 1: Large samples Test statistic/sampling distribution Critical values of the RR: z-scores Determining p-values with the z-table Assumptions Case 2: Small samples (equal variances assumed) Test statistic/sampling distribution Critical values of the RR: t-distribution with df = n1 + n2-2 Bracketing p-values with the t-table Assumptions - Test of Hypotheses about two population means involving matched pairs. Test statistic/sampling distribution Critical values of the RR: z or t values (depending on the sample size) Determining p-values (z-table and t-table) Assumptions

5 - Test of hypotheses about two population proportions Test statistic/sampling distribution Pooled sample proportion Critical values of the RR: z-table Computing p-values (z-table) Assumptions

6 Chapter 10: Analysis of Variance A. Completely Randomized Design - Introduction Cause-effect model Response (quantitative variable) Factor (categorical variable) Treatments: Factor levels (categories) Experimental units Total sample size n Experimental Design: Completely Randomized Statistical Model: One Way ANOVA - Hypothesis Testing Statistical Hypotheses (in symbols and words) Partition of the total variance Degrees of freedom Fundamental identities ANOVA table Test Statistic: formula and sampling distribution Use of the F table Rejection Region Assumptions Estimate of the common standard deviation Bracketing p-values with the F-table

7 B. Multiple Comparisons of Means - Two approaches: hypothesis testing and confidence intervals 1. Hypothesis Testing Hypotheses Test Statistic Rejection Region Multiple Comparison methods for the test statistic and critical values Interval estimator Decision rule 2. Confidence Intervals C. Randomized Block Design - Introduction Treatment Factor Block Factor Response: quantitative variable Treatments: levels of the Treatment factor Statistical design: Randomized Block Statistical model: Two-Way ANOVA (w/o Interaction) Assumptions

8 - Hypothesis Testing Partition of the total variance Degrees of freedom Fundamental identities ANOVA table Treatment effect and Block effect Statistical Hypotheses Test Statistics: formula and sampling distributions Rejection Region Assumptions Bracketing p-values with the F-table D. Two Factor Factorial Design - Introduction Factors: Two categorical variables A and B Response: quantitative variable Main effects Treatments: all combinations of Factor A and Factor B levels Replicates Statistical design: Two factor factorial Balanced factorial designs Statistical model: Two-Way ANOVA (with Interaction) Interaction effect: verbal and graphical interpretation Interpreting the table of sample means by treatment

9 - Inferences for Two-Way ANOVA with Interaction Basic elements Design: a x b Balanced Factorial Treatments: k = ( a )( b ) Replicates: r Total sample size: n = ( k )( r ) Partition of the total variance: Between and Within Treatments Between Treatment Sources: Factor A, Factor B, Interaction AB Degrees of freedom Fundamental identities ANOVA table Hypothesis Testing Treatment Effect Interaction Effect Separate Main Effects Conclusions

10 Chapter 11: Simple Linear Regression - Algebra Review Equations for straight lines Independent and dependent variables Slope and y-intercept Graphing straight lines Line patterns for different slopes - Probabilistic Model Explanatory and response variables Slope and y-intercept of the regression line Random error term. Assumptions - Least Squares (LS) Regression Least squares principle Computing and interpreting the LS estimates for the slope and y-intercept Graphing the LS regression line Using the LS regression equation for making predictions of the response variable. Extrapolations Prediction errors or residuals - Goodness of the fit of the LS Regression Line Interpreting: a) The estimated standard error of the model b) The coefficient of determination

11 - Statistical Inferences Confidence intervals for the slope and y-int Tests of hypotheses about the slope and y-int Test of significance for the regression equation Prediction intervals for the response variable - Linear Correlation between two random variables Coefficient of linear correlation Definition and interpretation Correlation vs. Causation Notation: parameter and statistic Range of values Graphical patterns Hypothesis Testing Statistical hypotheses in symbols and words Test Statistic Rejection Region: location and critical values

12 Chapter 13: Chi-Square Tests of Hypotheses A. Chi-square probability distribution Chi-square symbol Definition of the chi-square variable Parameter: degrees of freedom Graph of density curves: (a) n < 2 and (b) n > 2 Use of chi-square probability table B. Chi-square test for multinomial probabilities - Introduction Description of the problem One categorical variable: one way classification Statistical hypotheses: numerically and in words Observed and expected frequencies - Test Statistic Formula Observed and expected frequency Table format for computation Interpretation Sampling distribution: Chi-Square with df = k 1 where k is the number of categories - Assumptions 1) Units are an SRS 2) Units behave independent one to another 3) Sample size requirement: expected frequency for each category is at least five

13 - Rejection Region Size: determined by the significance level alpha Location: upper tail Critical values: Chi-Square table C. Chi-square test for the Independence/Dependence of two categorical variables - Introduction Two categorical variables: two-way classification Contingency tables: Row variable and Column variable Levels of the variables (r = no. of rows, c = no. of columns) Description of the problem - Hypothesis Testing Statistical hypotheses (in words) Sample data: contingency table Test Statistic Formula Observed and expected frequency Table format for computation Sampling distribution: Chi-square with df = (r - 1)(c - 1)

14 Assumptions 1) Units from all (row and column) levels are SRS 2) Units behave independent one to another 3) Expected frequency for each table cell is at least five Rejection Region Size: determined by the significance level alpha Location: upper tail (always) Critical Values: from the Chi-Square table

15 Chapter 14: Non-Parametric Statistics A. Wilcoxon Rank Sum Test Objective: comparing the location of center for two nonnormal populations Hypotheses Graphical representation of Ha Determining the ranking of sample data Test Statistic (W): rank sum for the smaller sample Rejection Region: determined by Ha and W Finding the critical values from Table XII Assumptions B. Kruskal-Wallis Test Objective: comparing the location of center for more than two non-normal populations Hypotheses Determining the ranking of sample data Test Statistic (H): based on the rank sums for all samples Rejection Region Finding the critical values from the Chi-Square table