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We use the estimate of the standard error to define the T-statistic: The estimate of the standard error is simply:
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As stated above (and developed in Chapter 8), the formula for the standard error is: Using these sample values, we can estimate the standard error of the distribution of sample means. The formula for the sample standard deviation is: Recall, from Chapter 4, that the sample variance and sample standard deviation are unbiased estimates of the population variance and populatin standard deviation. When we don't know the population's variability, we assume that the sample's variability is a good basis for estimating the population's variability. The limitation of Z-Tests is that we don't usually know the population standard deviation. Report for Univariate Analysis of Experiment 2 Data. Report for Univariate Analysis of Experiment 1 Data. Here is ViSta's report for these two experiments: We then look it up in the Z table to see how unusual the obtained sample's mean is, and decide if the null hypothesis Ho is true. We then can calculate Z by using the obtained sample mean. Note that the population mean is 18 under the null hypothesis, and the standard error is 1, as we just calculated. To determine how unusual the mean of the sample we will get is, we will use the Z formula to calculate Z for our sample mean under the assumption that the null hypothesis is true. The standard error is calculated by the formula:
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We continue to assume that population of normal rats has a mean of 18 grams with a standard deviation of 4. Lets assume that the researcher's sample has n=16 rat pups. We begin by reviewing the logic of hypothesis testing that underlies Z-Tests.Įxample: We return to the example concerning prenatal exposure to alcohol on birth weight in rats. They are based on T-Scores, which we first met in the lecture notes for Topic 4. These procedures use the T-Test, rather than the Z-Test. We don't usually know the population's standard deviation, which is required to compute the z-score's standard error.Ĭhapter 9 presents the statistical procedures that permit researchers to use a sample mean to test hypotheses about a population, when the population standard deviation is NOT known.