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The following 2x2 factorial data from problem 2 in section 4.3 show times for 2 fluids to drain from a container using tubes of 2 diameters. 4.50 4.00 Ln(Time) 3.50 3.00 Water 2.50 Glycol 2.00 1.50 0.15 0.2 0.25 Diameter 0.3 0.35 (a) Does there appear to be an interaction between fluids and diameters in the Ln scale? Why? The responses are not reasonably parallel indicating an important interaction. The p-value of 0.249 for the interaction effect in the ANOVA table below confirms this interaction. The fluid differences in the Ln scale are different for the smaller and larger diameter tubes with more of a fluid difference for the smaller diameter tube. (b) What does this interaction or lack of interaction tell us about consistency of span effect for the 2 materials in the original time scale? In the original scale the ratios of times for glycol:water would be larger for the smaller diameter tube. (c) Why would it not be useful to compare the main effect mean for 0.188” diameter values averaged over both liquids to the mean for 0.314” diameter values averaged over both liquids? The diameter effect is different for the two liquids. A combined effect wouldn’t match the effect in either liquid. We need to summarize diameter effects separately for water and glycol. Below is some info from Excel. There were 3 measurements in each treatment group for 18 total measurements. ANOVA Source of Variation Fluids Sample Diameter Columns Interaction Within Total SS 0.901 6.975 0.223 0.00819 df 1 1 1 8 8.107 11 MS 0.901 6.975 0.223 0.00102 (c) What is the value of the pooled standard variance S p2 ? The pooled standard variance is 0.00102. F 880.1 6815.9 217.9 P-value 1.81E-09 5.17E-13 4.36E-07 The means in the Ln scale are as shown below. Fluid Water Water Water Water Water Water Glycol Glycol Glycol Glycol Glycol Glycol Diam 0.188 0.188 0.188 0.314 0.314 0.314 0.188 0.188 0.188 0.314 0.314 0.314 Time 21.12 21.11 20.80 6.60 6.04 5.92 51.25 46.03 46.09 7.85 7.91 7.97 Mean St Dev CV SD/Mean 21.01 0.1819 0.0087 6.19 0.3630 0.0587 47.79 2.9966 0.0627 7.91 0.0600 0.0076 ln(Time) 3.050 3.050 3.035 1.887 1.798 1.778 3.937 3.829 3.831 2.061 2.068 2.076 Mean Similar to CV St Dev 3.045 0.0087 1.821 0.0579 3.866 0.0616 2.068 0.0076 (d) For water the difference between 0.188" and 0.314" means in Ln scale = 3.045 – 1.821 = 1.224. What is the corresponding ratio of times? 3.40 (e) Find a 2-sided 95% confidence interval for the difference in the Ln scale. Ln Scale Lower Limit 1.16 Ln Scale Upper Limit 1.28 (f) Find a 2-sided 95% confidence interval for the ratio of times to drain the container. Ratio Lower Limit 3.20 Ratio Upper Limit 3.61 According to problem 2 in section 4.3, a particular fluid model says that the time to drain should be inversely proportions to the 4th power of the diameters. What ratio of drain times would be predicted by this model? Is the confidence interval consistent with this ratio? Model ratio = 7.78. This ratio is clearly not in the confidence interval. We have very good evidence for rejecting the model.