z_and_t_tests - B Weaver(5-Oct-2007 z and t-tests 1...

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B. Weaver (5-Oct-2007) z- and t-tests ... 1 Hypothesis Testing Using z- and t-tests In hypothesis testing, one attempts to answer the following question: If the null hypothesis is assumed to be true, what is the probability of obtaining the observed result, or any more extreme result that is favourable to the alternative hypothesis? 1 In order to tackle this question, at least in the context of z- and t-tests, one must first understand two important concepts: 1) sampling distributions of statistics, and 2) the central limit theorem. Sampling Distributions Imagine drawing (with replacement) all possible samples of size n from a population, and for each sample, calculating a statistic--e.g., the sample mean. The frequency distribution of those sample means would be the sampling distribution of the mean (for samples of size n drawn from that particular population). Normally, one thinks of sampling from relatively large populations. But the concept of a sampling distribution can be illustrated with a small population. Suppose, for example, that our population consisted of the following 5 scores: 2, 3, 4, 5, and 6. The population mean = 4 , and the population standard deviation (dividing by N) = 1.414 . If we drew (with replacement) all possible samples of 2 from this population, we would end up with the 25 samples shown in Table 1. Table 1: All possible samples of n=2 from a population of 5 scores. First Second Sample First Second Sample Sample # Score Score Mean Sample # Score Score Mean 1 2 2 2 14 4 5 4.5 2 2 3 2.5 15 4 6 5 3 2 4 3 16 5 2 3.5 4 2 5 3.5 17 5 3 4 5 2 6 4 18 5 4 4.5 6 3 2 2.5 19 5 5 5 7 3 3 3 20 5 6 5.5 8 3 4 3.5 21 6 2 4 9 3 5 4 22 6 3 4.5 10 3 6 4.5 23 6 4 5 11 4 2 3 24 6 5 5.5 12 4 3 3.5 25 6 6 6 13 4 4 4 Mean of the sample means = 4.000 SD of the sample means = 1.000 (SD calculated with division by N) 1 That probability is called a p -value. It is a really a conditional probability --it is conditional on the null hypothesis being true.
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B. Weaver (5-Oct-2007) z- and t-tests ... 2 The 25 sample means from Table 1 are plotted below in Figure 1 (a histogram). This distribution of sample means is called the sampling distribution of the mean for samples of n=2 from the population of interest (i.e., our population of 5 scores). Figure 1: Sampling distribution of the mean for samples of n=2 from a population of N=5. I suspect the first thing you noticed about this figure is peaked in the middle, and symmetrical about the mean. This is an important characteristic of sampling distributions, and we will return to it in a moment. You may have also noticed that the standard deviation reported in the figure legend is 1.02, whereas I reported SD = 1.000 in Table 1. Why the discrepancy? Because I used the population SD formula (with division by N) to compute SD = 1.000 in Table 1, but SPSS used the sample SD formula (with division by n-1) when computing the SD it plotted alongside the histogram.
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