Cal Poly Pomona PSY 307 - Chapter 14 – t-Test for Two Independent Samples

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PSY 307 – Statistics for the Behavioral SciencesChapter 14 – t-Test for Two Independent SamplesIndependent Samples Observations in one sample are not paired on a one-to-one basis with observations in the other sample. Effect – any difference between two population means. Hypotheses: Null H0: 1–2= 0 ≤ 0 Alternative H1: 1–2≠ 0 > 0The Difference Between Two Sample MeansEffect SizeX1minus X2The null hypothesis (H0) is that these two means come from underlying populations with the same mean (so the difference between them is 0 and 1–2= 0).Sampling Distribution of Differences in Sample MeansCritical Value Critical ValueAll possible x1-x2difference scores that could occur by chance1–2Does our x1-x2exceed the critical value?x1-x2YES – reject the null (H0)What if the Difference is Smaller?Critical Value Critical ValueAll possible x1-x2difference scores that could occur by chance1–2Does our x1-x2exceed the critical value?x1-x2NO – retain the null (H0)Distribution of the Differences In a one-sample case, the mean of the sampling distribution is the population mean. In a two-sample case, the mean of the sampling distribution is the difference between the two population means. The standard deviation of the difference scores is the standard error of this distribution.Formulas for t-test (independent)21)()(2121xxhypsXXt221221nsnssppxx22121212nnSSSSdfSSSSsp121211)(nXXSS222222)(nXXSSEstimated standard errorEstimated Standard Error Pooled variance – the variance common to both populations is estimated by combining the variances. The variance average is computed by weighting the group variance by the degrees of freedom (df) then dividing by combined df. Df for pooled variance: n1+ n2- 2Confidence Intervals for t The confidence interval for two independent samples is: Find the appropriate value of t in the t table using the formula for df. The true difference in population means will lie between the upper and lower limits some % of the time))((2121 xxconfstXXAssumptions Both populations are normally distributed with equal variance. With equal sample sizes > 10, valid results will occur even with non-normal populations. Equate sample sizes to minimize effects of unequal variance. Increase sample size to minimize non-normality.Population Correlation Coefficient Two correlated variables are similar to a matched sample because in both cases, observations are paired. A population correlation coefficient ( ) would represent the mean of r’s for all possible pairs of samples. Hypotheses: H0: = 0 H1: ≠ 0t-Test for Rho ( ) Similar to a t–test for a single group. Tests whether the value of r is significantly different than what might occur by chance. Do the two variables vary together by accident or due to an underlying relationship?Formula for t212nrrthypStandard error of predictionCalculating t for Correlated Variables Except that r is used in place of X, the formula for calculating the t statistic is the same. The standard error of prediction is used in the denominator to calculate the standard deviation. Compare against the critical value for t with df = n – 2 (n = pairs).Importance of Sample Size Lower values of r become significant with greater sample sizes: As n increases, the critical value of t decreases, so it is easier to obtain a significant result. Cohen’s rule of thumb .10 = weak relationship .30 = moderate relationship .50 = strong


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Cal Poly Pomona PSY 307 - Chapter 14 – t-Test for Two Independent Samples

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