Bennett, PJ PSYCH 710 Hypothesis Testing Review of Hypothesis Testing PSYCH 710

Abstract

0.1 known population mean and variance Consider the following scenario: The population of a small town was unknowingly exposed to an environmental toxin over the course of several years. There is a possibility that exposure to the toxin during pregnancy adversely affects cognitive development which eventually leads to lower verbal IQ. To determine if this has happened to the children of this town, you use a standardized test to measure verbal intelligence in a random sample of 20 children who were exposed to the toxin. Based on extensive testing with typical children, the mean and standard deviation of the scores in the population of typical children is 100 and 10, respectively. The mean score of your sample was 93. Given this information, should we conclude that our sample of twenty scores was drawn from the population of typical scores? We will answer this question in what appears to be a roundabout way. We will start by assuming that there was no effect of the toxin, and therefore that our sample of scores was drawn from the population of typical scores. Hence, our null hypothesis is that the data were drawn from a population with a mean of 100 (µ = 100) and a standard deviation of 10 (σ = 10). Next, we have to evaluate whether our observation (i.e., the sample mean is 93) is unusual, or unlikely, given the assumption that the null hypothesis is true. If the observation is unlikely, then we reject the null hypothesis (µ = 100) in favor of the alternative hypothesis that µ ̸ = 100. In your textbook, the null and alternative hypotheses often are displayed thusly: H0: µ = 100 H1: µ ̸ = 100 How can we determine if our observation is unlikely under the assumption that the null hypothesis is true? Recall that the mean (µ ¯ Y) and standard deviation (σ ¯ Y) of the distribution of means – otherwise known as the sampling distribution of the mean – are related to the mean (µ) and standard deviation (σ) of the individual scores in the population by the equations where n is sample size. Therefore, for the current example µ ¯ Y = µ (1) σ ¯ Y = σ / √ n (2) µ ¯ Y = 100 (3) σ ¯ Y = 10 / √ 20 = 2.236 (4) Next, we note that if verbal IQ scores are distributed Normally, then the distribution of sample means will also be Normal. Moreover, the Central Limit Theorem implies the the distribution of sample means wil