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2. Sampling distribution of t • • The distribution of t values that would be obtained if a value of t were calculated for each sample mean for all possible random samples of a given size from a population t ratio: Sampling distribution of t • • • _ t= The result is a family of distributions, each associated with a specific degree of freedom Properties: • (X - µhyp) s • __ x Sampling distribution of t Sampling distribution of t • Degrees of freedom • • • Does the mean gas mileage for some population of cars drop below the legally required minimum of 45 miles per gallon? • • Sample: n = 6 cars _ X = 43; s = 2.19 the number of values free to vary, given one or more mathematical restrictions on a sample of observed values used to estimate some unknown population characteristic Degrees of freedom (single sample) • 3. Example: Gas mileage investigation Symmetrical, unimodal, bell-shaped (similar to the normal curve) For small values of df, tails are inflated, and the mean is smaller than the mean of a corresponding normal curve It gets closer and closer to a normal curve as the number of df increases df = n - 1 II. Inferential Statistics (8) ! Basics of Experimentation for a difference between means ! t test for two independent samples ! Testing 1 2. Testing for a difference between means 1.- Basics of Experimentation ! Experiment • independent & dependent variables ! Looking for cause-effect explanations ! The simplest case: Comparing two groups (samples) Two independent samples ! Control, • • Two related samples • • • Statistical Hypotheses • • • Nondirectional • • Directional (lower tail critical): H0: µ1 - µ2 ≥ 0 H1: µ1 - µ2 < 0 Directional (upper tail critical): H 0: µ 1 - µ 2 ≤ 0 H1: µ1 - µ2 > 0 • Testing for a difference between means • Standard error of X1 – X2 • A rough measure of the average amount by which any difference between sample means deviates from the difference between population means Any difference between two population means Testing for a difference between means H0: µ1 - µ2 = 0 H1: µ1 - µ2 ≠ 0 • Observations in one sample are paired, on a one-toone basis, with observations in the other sample Effect • Testing for a difference between means Observations in one sample are not paired, on a oneto-one basis, with observations in the other sample Sampling distribution of X1 – X2 • Differences between sample means based on all possible pairs of random samples (of given sizes) from two underlying populations 3. t test for two independent samples • • t ratio for two populations means (two independent samples) … as always: Statistic - Parameter Test statistic = Standard error of the statistic t= (X1 – X2) – (µ1 – µ2) s X1 – X2 2 t test for two independent samples • Estimated standard error of the difference between sample means t test for two independent samples • Example: see pdf file sX1 – X2 = √[s2 (1/n1 + 1/n2)] • Pooled estimate of variance (SS1 + SS2) s2p = n1 + n2 - 2 4. Other related concepts ! The ! p-values ! Statistical significance Effect Size ! Estimating ! Cohen’s d Statistical significance ! Tests p-values of hypotheses are often referred to as tests of significance. Test results are described as ! statistically significant if the H0 has been rejected ! not statistically significant if the H0 has been retained degree of rarity of a test result, given that the null hypothesis is true ! Smaller p-values tend to discredit the null hypothesis and to support the research hypothesis Statistical significance ! Implies only that H0 is probably false, and not whether it is false because of a large or small difference between population means (effect size) 3 Estimating Effect Size Estimating Effect Size: Cohen’s d ! Confidence intervals for µ1-µ2 of values that, in the long run, include the unknown effect a certain percent of the time ! Ranges ! CI for µ1-µ2 (two independent samples) ! X1-X2 ± (tconf)(sX1-X2) ! Spanish learning example (see PDF notes): -19.44 ± 2.13 * 5.5 ; [-31.16, -7.73] √s2p 19.44 19.44 = √128.07 11.32 = = 1.72 Estimating Effect Size: Cohen’s d ! Cohen’s guidelines for d Effect size is: ! small if d < 0.2 or d ≈ 0.2 ! medium if d ≈ 0.5 ! large if d > 0.8 or d ≈ 0.8 II. Inferential Statistics (9) 1.- Analysis of Variance ! Analysis ! ANOVA ! One of Variance way ANOVA ! Between-groups design ! Effect size ! Multiple comparisons X1-X2 st. deviation √s2p st. deviation supplies a stable frame of reference not influenced by increases in sample size d , the Standardized Effect Estimate, for the Spanish learning example (see PDF notes): = mean difference ! The ! Cohen’s |X1-X2| d , the Standardized Effect Estimate effect size by expressing the observed mean difference in standard deviation units ! Describes d= Estimating Effect Size: Cohen’s d d= ! Cohen’s ! An overall test of the null hypothesis for more than two population means ! Null Hypotheses ! H0 : µ0 = µ1 = µ2 ! One-way ANOVA ! The simplest type, that tests whether differences exist among population means categorized by only one factor or independent variable 4 Analysis of Variance (ANOVA) ! Two sources of variability ! Variability between groups • Variability among scores of subjects who, being in different groups, receive different experimental treatments ! Variability within groups • Variability among scores of subjects who, being in the same group, receive the same experimental treatments 5