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1、1CHAPTER 3: ANALYSIS OF VARIANCEOBJECTIVES:After completing this chapter, you should be able toDiscuss the general idea of analysis of variance.Use the one-way ANOVA technique to determine if there is a significant difference among two or more means.Use the two-way ANOVA technique to determine if th
2、ere is a significant difference in the main effects.3.1: INTRODUCTIONThe analysis of variance (ANOVA) is a procedure used to test whether difference exists between two or more population means for interval data.To do this, the technique analyzes the sample variance.The F test can also be used to tes
3、t the equality of two means. But since it is equivalent to the t test in this case, the t test is usually used instead of the F test when there are only two means. There are several reasons why the t test should not be done to test three means or more:21. When one is comparing two means at a time, t
4、he rest of the means under study are ignored. With the F test, all the means are compared simultaneously.2. When one is comparing two means at a time and making all pairwise comparisons, the probability of rejecting the null hypothesis when it is true is increased, since the more t tests that are co
5、nducted, the greater is the likelihood of getting significant different by chance alone.3. The more means there are to compare, the more t test are needed. (For example: For the comparison of 3 means two at a time, 3 t tests are required. For the comparison of 5 means two at a time, 10 tests are req
6、uired. And for the comparison of 10 means two at a time, 45 tests are required.)The F test can only show whether or not a difference exists but not where the difference lies.3F DISTRIBUTIONIf two independent samples are selected from two normally distributed populations in which the variance are equ
7、al (12 = 22 ) and if the variances s12 and s22 are compared as , the sampling distribution sof the ratio of the variances is called the F distribution.Characteristics of F distribution. F cannot be negative because variances are always positive or zero and it is a continuous distribution. The F dist
8、ribution is positively skewed. The F distribution is a family of curves based on the degrees of freedom of the variance of the numerator and the degrees of freedom of the variance of the denominator. Its values range from 0 to . As F , the curve approaches the X-axis but never touches it.Formula for
9、 the F test: 212swhere s12 is the larger of the two variances.4The F test has two terms for the degree of freedoms: that of the numerator, n1-1 and that of the denominator, n2-1, where n1 is the sample size from which the larger variance was obtained.3.2: ONE WAY ANALYSIS OF VARIANCEAssumption for t
10、he F test to compare three or more means:1. The populations from which the samples were obtained must be normally or approximately normally distributed.2. The samples must be independent of each other.3. The variances of the populations must be equal.Example 1:Fifteen fourth-grade students were rand
11、omly assign to three groups to experiment with three different methods of teaching arithmetic. At the end of the semester, the same test was given to all 15 students. The table gives the scores of students in the three groups.5In the F test, two different estimates of the population variance are mad
12、e.The first estimate is called the between-group variance, and it involves finding the variance of the means.The second estimate, the within-group variance, is made by computing the variance using all the data and it is not affected by differences in the means.If there is no difference in the means,
13、 the between group variance will be approximately equal to the within-group variance, and F test value will be close to 1 The null hypothesis will not be rejected.However, when the means differ significantly, the between-group variance will be much larger than the within-group variance, the F test w
14、ill be significantly greater than 1 The null hypothesis will be rejected.Method I Method II Method III48735165875585706990846895746761. Hypothesis StatementTo test the difference among the means, the following hypothesis should be used., where k is the number of groups012:.kHAt least two means diffe
15、r2. Test Statistic1SAMSkFEnTotal Sum of Squares, SST The total variability in the data. SST found using the following formula:222 21 or an ijij ijij xSTxXSTn7 SST can be partitioned into 2 component parts: the treatment sum of square, SSA and the error sum of square, SSE.SST=SSA + SSETreatment Sum o
16、f Squares, SSA The sum of squares between groups.2221 or a ijiiii i xxSAnXSAnn Error Sum of Squares, SSE Sum of squares within groups.2221s or i ii ij ixSEnSExnor SSE = SST SSAMean Squares The mean squares values are equal to the sum of squares divided by the degree of freedom. and 1SASEMMk nk8ANOVA Summary TableSource Sum of df Mean FSquares SquaresTreatmen
