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Example 12.2 A chemical process is being investigated in a pilot plant. The factors under study are, first the catalyst, Liquid Catalyst 1 or Liquid Catalyst 2. and then the concentration of each (1 gram/liter or 2 grams/liter). Two replicate runs are done for each combina- tion of factors. The response, or dependent variable, is the percentage yield of the desired product. Results are shown in Table 12.6 below. Table 12.6: Percentage Yields for Catalyst Study Yield Catalyst Catalyst Concentration 1 2 1g/L 49.3 47.4 53.4 50.1 2g/L 63.6 49.7 59.2 49.9 308 Introduction to Analysis of Variance Is the yield significantly different for a different catalyst or concentration or combina- tion of the two? Use the 5% level of significance.Example 5.12 The number of meteors found by a radar system in any 30-second interval under specified conditions averages 1.81. Assume the meteors appear randomly and inde- pendently. a) What is the probability that no meteors are found in a one-minute interval? b) What is the probability of observing at least five but not more than eight meteors in two minutes of observation?Example 6.3 The random variable of Example 6.1 has the probability density function given by: A(x) = 0 for x 2 a) Find the probability that X is between 1 and 2. b) Find the cumulative distribution function of X. c) Find the expected value of X. d) Find the variance and standard deviation of X.Example 9.10 We decide to run a test using an experimental evaporation pan and a standard evaporation pan over ten successive days. The two types are set up side-by-side so 238 Statistical Inferences for the Mean that atmospheric conditions should be the same. A coin is tossed to decide which evaporation pan is on the lefthand side and which on the righthand side on any particular day. The measured daily evaporations are as follows: Pair or Day No. 1 2 3 4 5 6 7 8 9 10 Evaporation, mm: Pan A 9.1 4.6 14.0 16.9 11.4 10.7 27.4 22.8 42.8 29.4 Pan B 6.7 3.1 13.8 16.6 12.3 6.5 24.2 20.1 41.9 27.7 d = Agym. A - B 2.4 1.5 0.2 0.3 -0.9 4.2 3.2 2.7 0.9 1.7 Does the experimental Pan A give significantly higher evaporation than the standard Pan B at the 1% level of significance?Example 9.7 The electrical resistances of components are measured as they are produced. A sample of six items gives a sample mean of 2.62 ohms and a sample standard devia- tion of 0.121 ohms. At what observed level of significance is this sample mean significantly different from a population mean of 2.80 ohms? Is there less than 2% probability of getting a sample mean this far away from 2.80 ohms or farther purely by chance when the population mean is 2.80 ohms?Example 10.3 Two additives to Portland cement are being tested for their effect on the strength of concrete. 21 batches were made with Additive A, and their strengths showed standard deviation s, = 41.3. 16 batches were made with the same percentage of Additive B. and their strengths showed standard deviation s, = 26.2. Assume that the strengths of concrete follow a normal distribution. Is there evidence at the 1% level of signifi- cance that the concrete made with Additive A is more variable than concrete made with Additive B