A researcher conducted an experiment to test the effects of running 3 different types of gasoline with 3 possible types of additive on the mileage obtained. The experiment assumed that there is an interaction between gasoline type and the type of additive, hence, a two-factor ANOVA was employed in the data analysis. It was decided to run 36 motors – 4 in each factor level combinations. The following are the analysis results MEAN   ADDITIVE 1 ADDITIVE 2 ADDITIVE 3   GAS TYPE 1 125.825  130.775 128.275 128.2917 GAS TYPE 2 127 133.6 137.825 132.8083 GAS TYPE 3 126.35 132.5 123 127.2833   126.3917 132.2917 129.7 129.4611 Analysis of Variance Table   Df Sum Sq Mean Sq F Value Pr(>F) factor(gas.type) 2 207.7706 103.8853 11.08482 0.000306 factor(additive)  2 209.8872 104.9436 11.19775 0.000287 factor(gas.type):factor(additive)  4 262.9678 65.74194 7.01483 0.000523 Residuals 27 253.04 9.371852 NA NA a.) Is there a significant interaction between the type of gasoline and the type of additive? Why? b.) What does the result on the test for significant interaction imply? c.) Say you are to choose between the three gasoline types, regardless of the type of additive used and considering all other factors constant. Which gasoline would you prefer for your motor and why?

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...