PSYCH 240 1st Edition Lecture 13Section 7: Central Limit Theorem cont.Advantages of Bigger Sample Sizes-The larger your sample size, the more likely you are to find specific evidence for the alternative hypothesis if it is true-Larger sample sizes produce less variable comparison distributions, and having a less variable distribution to compare to makes it easier to figure out that your sample is atypical. Sample Size-Example:oOn a test for cognitive ability, scores for children with Down Syndrome are normally distributed with a mean of 50 and a sd of 15. a researcher develops an intensive early-learning program that lasts from ages 1-6, and the children in the program take the test at age 7.oScenario 1: 1 child completes the program and scores 57 on the testoScenario 2: 40 children completed the program and had an average score of 57 on the testoDid the program improve test scores?---The critical value is closer to the mean of the comparison distribution with 40 scores and with 1 score because the distribution for 40 scores is less variable-A sample value of 57 would produce a non-significant test if it was the value ofa single score-A sample value of 57 would produce a significant test if it was a mean of 40 scores-Common objection to hypothesis testing is not accurateo"You can make anything significant if you run enough subjects"oIf the null hypothesis is true, then the probability of getting a significant result stays equal to alpha regardless of sample size-Three types of means:oComparison Mean: the population mean of the comparison distributionoTrue Mean: the mean of the population from which your data were sampledoSample Mean: the one mean for the one sample you happened to get for your study Clicker Questions-What does "specific" mean with regard to the evidence for a hypothesis?oIf the hypothesis is false, then the evidence is unlikely to be observed-What does a significant result mean for a null hypothesis significance test?oYou got a result that probably would NOT have occurred if the null hypothesis were true-The comparison distribution shows us what we can expect to see under…oThe null hypothesis-True or false: If the alternative hypothesis is true, then you know that you have a high probability of getting a significant result.oFalse -True of false: The null hypothesis always states that a sample collected by a research belongs to a known comparison population, and the alternative hypothesis says that it does not belong to that populationoTrue-We conduct a hypothesis test with alpha = .05. What is the probability of getting a significant result if the null hypothesis is true?o.05-We conduct a hypothesis test with alpha = .05. What is the probability of getting a non-significant result if the null hypothesis is true?o.95-We conduct a hypothesis test with alpha - .05. What is the probability of getting a significant result if the alternative hypothesis is true?oCould be anything higher than .05 - other than that we don't know-We conduct a hypothesis test with alpha - .05. What is the probability of getting a non-significant result if the alternative hypothesis is true?oCould be anything lower than .95 - other than that we don't know-A population of scores has a mean of 20 and a standard deviation of 24. A researcher takes a sample of 16 scores from this population, and the sample mean is 14. What is the z-score for the sample mean on the sampling distribution of the mean?oSample mean (M):14oStandard deviation(σ): 24oMean of the sampling distribution of the mean(μM): 20oStandard deviation of the sampling distribution of the mean(σM): 2oNumber of scores(N): 16oZ-score(Z): -1- - Objectives-Know the central limit theorem. Understand how the distribution of means (a.k.a. the sampling distribution of the mean) it is constructed in theory. -Be able to determine the mean and variability of the distribution of means given the mean and variability of the distribution of scores and the sample size. -Understand how the shape and variability of the distribution of means relates to the shape and variability of the distribution of scores, and how this relationship changes with different sample sizes. -Be able to explain in words why the distribution of means becomes normal for large sample sizes even if the distribution of scores is not normal. -Be able to explain in words why the variance of the distribution of means decreases as sample size increases.-Be able to compute the standard error of the mean (SEM; that is, the standard deviation of the distribution of means) given the sample size and the variability of the distribution ofscores. -Know that the error bars reported in research articles often report the SEM. -Be able to explain why hypothesis testing is more effective with larger sample sizes. -For hypothesis tests, know how the probability of getting a significant result changes when sample size increases for both cases in which the alternative hypothesis is true and cases in which the null hypothesis is true. Understand why increasing the sample size has adifferent effect based on which hypothesis is true. -Given the mean and standard deviation of the comparison distribution of scores and the sample size, be able to compute the z-score of a sample score relative to the distribution of scores, and be able to compute a z-score for a sample mean relative to the distribution of