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Chapter Twelve PSY201

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Kristie Dukewich

Chapter Twelve: Sampling Distributions, Sampling Distribution of the Mean, the Normal Deviate (z) Test What is the probability of getting the obtained result or results even more extreme if chance alone is responsible for the differences between the experimental and control scores? + two steps involved + 1. calculating the appropriate statistic + 2. evaluating the statistic based on its sampling distribution Sampling Distributions sampling distribution of a statistic gives (1) all the values that the statistic can take and (2) the probability of getting each value under the assumption that it resulted from chance alone in replicated measures design, used sign test to analyze the data + statistic calculated was number of pluses in sample of N difference scores + in marijuana and appetite experiment, obtained 9 pluses and 1 minus result was evaluated using binomial distribution + binomial distribution with P = 0.50 lists all possible values of the statistic, number of pluses, along with probability of getting each value under assumption that chance alone produced it + binomial distribution with P = 0.50 is the sampling distribution of the statistic used in the sign test . There is a different sampling distribution for each sample size (N) data analysis involves two steps + 1. calculating appropriate statistic number of pluses and minuses for the sign test (example) + 2. evaluating the statistic based on its sampling distribution if probability of getting obtained value of statistic or any value more extreme is equal to or less than alpha level, reject H0 and accept H1 if not, retain H0. + if reject H0 and it's true Type I error + retain H0 and it's false Type II error + process applies to all experiment involving hypothesis testing + what changes from experiment to experiment is the statistic used and its accompany sampling distribution Generating Sampling Distributions defined sampling distribution as a probability distribution of all possible values of a statistic under the assumption that chance alone is operating + one way of deriving sampling distributions is from basic probability considerations + sampling distributions can also be derived from an empirical sampling approach have an actual or theoretical set of population scores that exists if the independent variable has no effect derive the sampling distribution of statistic by + 1. determining all possible different samples of size N that can be formed from population of scores + 2. calculating the statistic for each of the samples + 3. calculating the probability of each value of the statistic if chance alone is operating ie: conducting an experiment with sample size N = 2, using sign test for analysis + imagine a theoretical set of scores that'd result if experiment were done on entire population and independent variable had no effect + population set of scores null-hypothesis population null-hypothesis population is an actual or theoretical set of population scores that would result if the experiment were done on the entire population and the independent variable had no effect. Called this because it's used to test the validity of the null hypothesis in case of sign test, if independent variable had no effect, null hypothesis population would have an equal number of pluses and minuses (P = Q = 0.50) for computational ease in generating sampling distribution, assume there's only six scores in population: three pluses and three minuses + to derive sampling distribution of number of pluses with N = 2, must determine all different samples of size N that can be formed from population + sampling is one at a time, with replacement + Figure 12.1 show population and different samples of size 2 that can be drawn from it there are 36 different samples of size 2 possible (column 2) + next, calculate the value of the statistic for each sample information presented in table of 12.1, columns 3 and 4 + of 36 different samples possible, 9 have two pluses, 18 have 1 plus and 9 have 0 plus + calculate the probability of getting each value of statistic. If chance alone is operating, each sample is equally likely we now derived the sampling distribution for N = 2 of statistic number of pluses + distribution plotted in Figure 12.2 + used a population in which there were only six scores + identical sampling distribution would've resulted (there'd be many more different samples) had we used a larger population as long as number of pluses equaled the number of minuses and the sample size equaled 2 same sampling distribution arrived at through basic probability considerations when discussing binomial distribution with N = 2 + this time, generated it by sampling from null-hypothesis population. sampling distribution gives all values a statistic can take, along with probability of getting each value if sampling is random from null-hypothesis population The Normal Deviate (z) Test a test that's used to know when we know the parameters of the null-hypothesis population + uses the mean of the sample as a basic statistic no doubt that sample mean of 72 is lower than national population mean of 75 + if chance alone is at work, then we can consider 100 sample scores to be a random sample from a population with u = 75 + null-hypothesis for this experiment asserts that such is the case what's the probability of getting a mean score as low as or even lower than 72 if 100 scores are a random sample from a normally distributed population having a mean of 75 and standard deviation of 16? + if probability is equal to or lower than alpha, reject H0 and accept H1
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