An experiment to study the effects of the presence of four different sugar

Example 10.1

3. An experiment to investigate whether hardwood concentration in pulp

has an effect on tensile strength of bags made from the pulp

4. An experiment to decide whether the color density of fabric specimens

depends on the amount of dye used In 1 the factor of interest is gasoline brand, and there are five different levels of the factor. In 2 the factor is sugar, with four levels or five, if a con- trol solution containing no sugar is used. In both 1 and 2, the factor is qual- itative in nature, and the levels correspond to possible categories of the factor. In 3 and 4, the factors are concentration of hardwood and amount of dye, respectively; both these factors are quantitative in nature, so the levels identify different settings of the factor. When the factor of interest is quantitative, sta- tistical techniques from regression analysis discussed in Chapters 12 and 13 can also be used to analyze the data. This chapter focuses on single-factor ANOVA. Section 10.1 presents the F test for testing the null hypothesis that the population or treatment means are identical. Section 10.2 considers further analysis of the data when H has been rejected. Section 10.3 covers some other aspects of single-factor ANOVA. Chapter 11 introduces ANOVA experiments involving more than a single factor. 10.1 Single-Factor ANOVA Single-factor ANOVA focuses on a comparison of more than two population or treat- ment means. Let the number of populations or treatments being compared the mean of population 1 or the true average response when treatment 1 is applied the mean of population I or the true average response when treatment I is applied The relevant hypotheses are versus If , H is true only if all four m i ’s are identical. H a would be true, for example, if , if , or if all four m i ’s differ from one another. A test of these hypotheses requires that we have available a random sample from each population or treatment. The article “Compression of Single-Wall Corrugated Shipping Containers Using Fixed and Floating Test Platens” J. Testing and Evaluation, 1992: 318–320 describes an experiment in which several different types of boxes were compared m 1 5 m 3 5 m 4 2 m 2 m 1 5 m 2 2 m 3 5 m 4 I 5 4 H a : at least two the of the m i ’s are different H : m 1 5 m 2 5 c5 m I m I 5 m 1 5 I 5 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Table 10.1 The Data and Summary Quantities for Example 10.1 Type of Box Compression Strength lb Sample Mean Sample SD 1 655.5 788.3 734.3 721.4 679.1 699.4 713.00 46.55 2 789.2 772.5 786.9 686.1 732.1 774.8 756.93 40.34 3 737.1 639.0 696.3 671.7 717.2 727.1 698.07 37.20 4 535.1 628.7 542.4 559.0 586.9 520.0 562.02 39.87 Grand mean ⫽ 682.50 630 4 3 2 1 660 690 750 720 780 550 4 3 2 1 600 650 a b 700 750 Figure 10.1 Boxplots for Example 10.1: a original data; b altered data With m i denoting the true average compression strength for boxes of type i , 2, 3, 4, the null hypothesis is . Figure 10.1a shows a com- parative boxplot for the four samples. There is a substantial amount of overlap among observations on the first three types of boxes, but compression strengths for the fourth type appear considerably smaller than for the other types. This suggests that H is not true. The comparative boxplot in Figure 10.1b is based on adding 120 to each obser- vation in the fourth sample giving mean 682.02 and the same standard deviation and leaving the other observations unaltered. It is no longer obvious whether H is true or false. In situations such as this, we need a formal test procedure. H : m 1 5 m 2 5 m 3 5 m 4 i 5 1 with respect to compression strength lb. Table 10.1 presents the results of a single- factor ANOVA experiment involving types of boxes the sample means and standard deviations are in good agreement with values given in the article. I 5 4 ■ Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.