1. Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Chapter 6 Hypothesis Tests with Means of Samples

2. Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall The Distribution of Means The distribution of means is –A distribution of the means of a large number of samples of the same size –Each sample randomly selected from the same population of individuals

3. Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Characteristics of the Distribution of Means Mean –Same as the mean of the population of individuals Variance –Variance of the population of individuals divided by the number of individuals in each sample –SD of the distribution of means is the square root of the variance of the distribution of means Shape –Approximately normal if either Each sample as at least 30 observations Underlying distribution of the population of individuals is normal –In general, unimodal, symmetrical, less spread out than the distribution of the population of individuals

4. Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Review of Three Kinds of Distributions We’ve considered three kinds of distributions A.The distribution of the population of individuals B.The distribution of a sample C.The distribution of means of samples

5. Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Hypothesis Testing with a Distribution of Means Figuring Z score of a sample’s mean on the distribution of means –Conceptually the same as figuring a Z score for an individual on a distribution of a population –Subtract the mean of the distribution of means from the sample’s mean –Then divide by the SD of the distribution of means

6. Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall First Example: The first-year students of Shandong Normal University had a test of computer literacy (The total score is 150). The mean of the score was 100 (μ=100). The standard deviation was 15 (s=15). There were 70 students in educational technology department. They got the mean of the score was 103.3. Could we get the conclusion that the performance of students in educational technology department was higher than general first-year students of Shandong Normal University?

7. Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

8. Estimation and Confidence Intervals Point estimate –Estimate of a specific value of a population parameter (e.g., the mean) –The best estimate of the mean of a population is the mean of a sample taken from that population –SD of the distribution of means (called the “standard error”) is a measure of the accuracy of the point estimate

9. Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Estimation and Confidence Intervals Confidence interval –Interval estimate with a given likelihood of including the population mean –Typical confidence intervals 95% 99% –The larger the interval, the greater one’s confidence than the interval contains the true mean

10. Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Steps for Figuring 95% and 99% Confidence Intervals Estimate population –Best estimate is sample mean Figure SD of distribution of means Find Z scores that go with confidence limits you want –For 95%, Z scores are +1.96 and -1.96 – For 99%, Z scores are +2.57 and - 2.57 Convert Z scores to raw scores –Multiply Z score by SD of distribution of means and add to the population mean

11. Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Hypothesis Testing Confidence intervals sometimes used instead of, or alongside, ordinary hypothesis testing –If confidence interval does not include the mean of the null hypothesis distribution, result is statistically significant Rare for researchers to know the characteristics of the population distribution, but when they do, the hypothesis test that is appropriate is a Z test.

12. Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Second Example: The standard deviation of computer literacy performance at Shandong Normal University was 5. There was a class which included 49 students at Shandong Normal University. The average score of this class was 85. Please inferred the confidence interval of computer literacy performance about the students at Shandong Normal University.

13. Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall