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CHAPTER 13 – STATISTICAL INFERENCE: TWO SAMPLES 13.1 Introduction to Hypothesis Tests for 2 Samples Examples at the beginning of the chapter involve a comparison of 2 population distributions (example: are men able to withstand weightlessness better than women?, etc.) Population distributions can differ in central tendency, dispersion, skewness, and kurtosis. Our inferences in this chapter are just extensions of the 1 sample case you learned in chapters 10-12 13.2 Two-Sample t Test and Confidence Interval for Using Independent Samples A t test statistic is used to test a hypothesis about the means, and , of 2 populations The statistic can be used to test any of the following null hypotheses: is the hypothesized difference between the population means Usually we are interested in testing whether the 2 population means are equal, in which case . The t test statistic is given by:
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The denominator of the t statistic, , is an estimator of the standard error of the difference between 2 population means. Degrees of freedom for the t statistic: Sample 1 contributes degrees of freedom, the number of d.f. associated with Sample 2 contributes degrees of freedom, the number of d.f. associated with The null hypothesis is rejected if the observed t statistic exceeds or equals the critical value of t given in Appendix Table D.3. 1 and 2 tailed critical values for t are t α,ν and t α/2,ν To use the t statistic, assume that 2 random samples of sizes n 1 and n 2 have been obtained from the populations of interest or that participants have been randomly assigned to 2 groups often called experimental and control groups. These sampling procedures produce independent samples in which the selection of elements in 1 sample is not affected by the selection of elements in the other. The use of random sampling or random assignment helps to ensure that the samples are statistically independent. Random assignment helps to distribute the unique characteristics of the participants equally to the 2 groups. Assume that the 2 populations are normally distributed and that the variances of the populations and are unknown but assumed to be equal. The pooled variance in the t formula is used since we are assuming the unknown population variances, and , are equal. If the equality assumption is tenable, the sample variances, and , are both estimators of the same population variance, . Whenever 2 estimators of are available, a pooled estimator is likely to provide a better estimate of the unknown population variance than either of the sample estimators taken alone. The pooled variance is just a weighted mean of and where the weights are the respective degrees of freedom. Homogeneity of variance assumption – the assumption that the variances of populations 1 and 2 are equal When we expose 1 group to a treatment and use the other group as the control, we often expect the treatment to raise or lower the scores of those exposed to it by a constant amount. Remember that adding or subtracting a constant (4.2 exercise 8 & 4.7 exercise 11) does not affect the standard deviation (or variance) of the scores.
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