ME495+Statistics+Homework+Solutions

ME495+Statistics+Homework+Solutions - ME495 Statistics...

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ME495 Statistics Homework Solutions 1. Single-factor ANOVA In an experiment to compare the quality of four different brands of recording tape, five reels of each brand were selected and the number of flaws in each reel was determined. (source: Jay L. Devore, Probability and Statistics for Engineering and Sciences , Wadsworth, Inc., 1982, p. 370) Is the expected number of flaws per reel the same for each brand? brand A 10 5 12 14 8 B 14 12 17 9 8 C 13 18 10 15 18 D 17 16 12 22 14 Solution: Create a matrix flaw in Matlab. The first row represents the data for brand A, the second row for brand B, the third row for brand C and the fourth row for brand D. Then use the function anova1 to perform a one-way ANOVA (i.e., single-factor ANOVA): >>[p, tbl, stats] = anova1(flaw’) The resulting ANOVA table looks like this: The reason to use the transpose of flaw ( flaw’ ) is that each column (rather than row) represents an independent sample containing different observations. The result p is the probability of the null hypothesis that all samples are drawn from the same population (or from different populations with the same means). If the p-value is near zero, this casts doubt on the null hypothesis and suggests that at least one sample mean is significantly different than the other sample means. It is common to declare a result significant if the p-value is less than 0.05. In this example, since p-value is greater than 0.05, we conclude that the sample means are barely the same. The result tbl returns the ANOVA table in a cell array. This table is also automatically plotted in a separate figure. The ANOVA table has columns for the sums of squares (SS), degree of freedom (df), mean squares (MS), F statistic, and p-value. In this case the p-value is about
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0.0529. An F statistic as extreme as the observed F would occur by chance once in less than twenty times if the number of flaws were truly equal. The result stats returns a stats structure that can be used to perform a follow-up multiple comparison test. 2. Two-factor ANOVA analysis A study to determine the effect of car model and factory on the gas mileage of cars shows the following results (the unit has been converted from miles/gallon to kilometers/liter; source: Matlab Statistics Toolbox User’s Guide Version 5 , the Mathworks, Inc., p. 133). There are three models of cars (columns) and two factories (rows). The reason there are six rows
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ME495+Statistics+Homework+Solutions - ME495 Statistics...

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