Sampling distribution of t. 2. Sampling distribution of t. 3. Example: Gas mileage investigation. II. Inferential Statistics (8) t =

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1 2. The distribution of t values that would be obtained if a value of t were calculated for each sample mean for all possible random of a given size from a population _ t ratio: (X - µ hyp ) t s x The result is a family of distributions, each associated with a specific degree of freedom Properties: 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 Degrees of freedom 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) df n Example: Gas mileage investigation 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 II. Inferential Statistics (8)! Basics of Experimentation!! t test for two independent 1

2 1.- Basics of Experimentation! Experiment! Control, independent & dependent variables! Looking for cause-effect explanations! The simplest case: Comparing two groups () 2. Testing for a difference between Two independent Observations in one sample are not paired, on a oneto-one basis, with observations in the other sample Two related Observations in one sample are paired, on a one-toone basis, with observations in the other sample Effect Any difference between two population Statistical Hypotheses Nondirectional H 0 : µ 1 - µ 2 0 H 1 : µ 1 - µ 2 0 Directional (lower tail critical): H 0 : µ 1 - µ 2 0 H 1 : µ 1 - µ 2 < 0 Sampling distribution of X 1 X 2 Differences between sample based on all possible pairs of random (of given sizes) from two underlying populations Directional (upper tail critical): H 0 : µ 1 - µ 2 0 H 1 : µ 1 - µ 2 > 0 Standard error of X 1 X 2 A rough measure of the average amount by which any difference between sample deviates from the difference between population 3. t test for two independent t ratio for two populations (two independent ) as always: Test statistic Statistic - Parameter Standard error of the statistic t (X 1 X 2 ) (µ 1 µ 2 ) s X1 X 2 2

3 t test for two independent Estimated standard error of the difference between sample t test for two independent Example: see pdf file s X1 X 2 [s 2 (1/n 1 + 1/n 2 )] Pooled estimate of variance s 2 p (SS 1 + SS 2 ) n 1 + n Other related concepts p-values! p-values! Statistical significance! Estimating Effect Size! Cohen s d! The 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! Tests of hypotheses are often referred to as tests of significance. Test results are described as! statistically significant if the H 0 has been rejected! not statistically significant if the H 0 has been retained Statistical significance! Implies only that H 0 is probably false, and not whether it is false because of a large or small difference between population (effect size) 3

4 Estimating Effect Size! Confidence intervals for µ 1 -µ 2! Ranges of values that, in the long run, include the unknown effect a certain percent of the time! CI for µ 1 -µ 2 (two independent )! X 1 -X 2 ± (t conf )(s X1-X2 )! Spanish learning example (see PDF notes): ± 2.13 * 5.5 ; [-31.16, -7.73]! Cohen s d, the Standardized Effect Estimate! Describes effect size by expressing the observed mean difference in standard deviation units d mean difference st. deviation X 1 -X 2 s 2 p! The st. deviation supplies a stable frame of reference not influenced by increases in sample size d! Cohen s d, the Standardized Effect Estimate, for the Spanish learning example (see PDF notes): X 1 -X 2 s 2 p ! 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)! Analysis of Variance! One way ANOVA! Between-groups design! Effect size! Multiple comparisons 1.- Analysis of Variance! ANOVA! An overall test of the null hypothesis for more than two population! Null Hypotheses! H 0 : µ 0 µ 1 µ 2! One-way ANOVA! The simplest type, that tests whether differences exist among population categorized by only one factor or independent variable 4

5 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