Cumulative Final Exam Review: Key Concepts in Statistics

The Final Exam is Monday 12/11 in our normal classroom from 7:30am-9:30am. It will be cumulative in coverage. You are allowed to use one sheet (front and back) of hand-written notes. You may use a calculator. Make sure you have pencils, paper and a way to scan and upload your exam. You will not have access to computer software during the Exam. However, questions surrounding sorting, inputting and processing data using Excel and JMP Pro are fair game. Concept Inventory: 1. Basic Statistics in Investigations a. Population parameters and sample statistics. b. Selecting an appropriate statistical test. i. Z-score ii. Student’s t-distribution iii. F-distribution iv. One or two factor ANOVA v. Factorial Design c. Characterizing and classifying experimental error. i. Mean ii. standard deviation iii. variance iv. pooled standard deviation v. standard error d. Replication, randomization and blocking. e. Normal distribution and the central limit theorem 2. Hypothesis Testing a. Null and Alternative hypotheses b. confidence intervals 3. Factorial Designs a. Model Selection for multiple factors b. Interpretation of DOE model parameters and sketch the model. c. Interpretation of JMP DOE model report (ie. Significance, regression coefficients and model fit to data.) 4. Model Validation a. RMSE b. CV c. Grubb’s test for outliers

Practice Final Exam ECH3844 Chemical Engineering Statistics **This will not be submitted for credit. A solution will NOT be posted. You can ask questions during Dr. Hunter’s office hours. The length and difficulty of the following practice exam is similar to the final. Concepts in the list above may appear on the final exam, even if they are not covered in the questions below. 1. Select the appropriate statistical test to provide evidence to address the following questions for scientific investigation. Not all choices will be used, and some may be used more than once.: _______ _______ _______ _______ _______ _______ Is measured value 𝑦𝑦 0 is part of a normally distributed population, with mean, 𝜇𝜇 , and standard deviation, 𝜎𝜎 ? Does a change in a single factor with 5 possible levels have an effect on the value of the response variable? Is there a correlation between two or more factors and response variable, which can be reported as a statistical regression model? Does a change in a single factor from level 1 to level 2 have an effect on the value of the response variable; n 1 samples are collected at factor level 1 and n 2 samples are collected at factor level 2? Does a change in factor from level 1 to level 2 have an effect on the value of the response variable; also, does a secondary factor influence the response? Does a change in a single factor from level 1 to level 2 have an effect on the variance of the response variable; n 1 samples are collected at factor level 1 and n 2 samples are collected at factor level 2? a. z -score b. Student’s t test c. chi-squared test d. F statistic e. one-way ANOVA f. two-way ANOVA g. DOE analysis

2. Consider the following set of data: 173 179 199 178 198 182 190 a. Calculate the mean, variance, standard deviation, degrees of freedom and standard error. 𝑦𝑦 � = ___________ 𝑠𝑠 2 = ___________ 𝑠𝑠 = ___________ 𝜈𝜈 = ___________ 𝑠𝑠𝑠𝑠 = ___________ b. Determine the critical t value and a 95% confidence interval 𝑡𝑡 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = ___________ 𝐶𝐶𝐶𝐶 : 𝑙𝑙𝑙𝑙𝑙𝑙𝑠𝑠𝑙𝑙 = __________ to 𝑢𝑢𝑢𝑢𝑢𝑢𝑠𝑠𝑙𝑙 = ___________ c. Find the t-statistic corresponding to a value of 200. Use α =0.05, to make a claim as to whether the value 200 is likely to be part of the normally distributed population. 𝑡𝑡 0 = ___________

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3. You have been hired as an intern at the FAMSTATE Adhesive Co. Your first assignment is to design an experiment to determine how the application pressure and temperature affect the shear strength (lb/in 2 ) of the adhesive. A range of temperatures that span normal atmospheric conditions (273 K to 310 K) and a range of pressures that represent the typical application of force that could be exerted by a person (10 to 100 lb/in 2 ) should be investigated. Assume that this experiment is relatively inexpensive to perform. a. You are asked to develop an empirical model that can be used to predict shear strength under varying application conditions, where the model form assumes that pressure has a linear effect and temperature has a non-linear effect on shear strength. Write out the general form of the DOE model (hint: use b-parameters, not value to represent the effects). 𝑦𝑦 � = _____________________________________________________________________________________ b. Define the response variable with units, and factor levels in the following table: Response ( 𝒚𝒚 � ) - 0 + Factor 1 (X 1 ) Pressure (lb/in 2 ) Factor 2 (X 2 ) Temperature (K) c. How many trials would be required to conduct a factorial design experiment with the minimum number of replicates to identify outliers?

4. In your new role as a Quality Engineer, you study the variability of a continuously stirred tank reactor (CSTR) used to produce the adhesive. In order to set product specifications, you must characterize the location and spread of the primary response variable, the fractional conversion of the limiting reactant, X . You run 5 experimental trials, all under the same, standard operating conditions and measure the steady state fractional conversion, X . Data: 0.45 0.51 0.67 0.47 0.44 Statistics: 𝑦𝑦 � = 0.508 𝑠𝑠 = 0.0845 When you survey the data, you observe that the third trial fractional conversion seems suspiciously high. Uses Grubb’s test to determine whether this suspiciously high data point is an outlier that should be excluded from analysis. Use α =0.05. 𝐺𝐺 = ___________ 𝑡𝑡 𝛼𝛼 2𝑛𝑛 ⁄ , 𝑛𝑛−2 = ___________ 𝐺𝐺 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = ___________ Claim:

5. A published study of a chemical reaction, 𝐴𝐴 → 𝑃𝑃 , indicates that if the reactor initially contains A at a concentration 𝐶𝐶 𝐴𝐴0 ( 𝑔𝑔 𝐿𝐿 ) and the reaction temperature, T , is kept constant, then the concentrations of P in the reactor increases with time according to the formula 𝐶𝐶 𝑝𝑝 � 𝑔𝑔 𝐿𝐿 � = 𝐶𝐶 𝐴𝐴0 (1 − 𝑠𝑠 −𝑘𝑘𝑐𝑐 ) The rate constant, 𝑘𝑘 = 0.011 ( 𝑠𝑠 −1 ) , is assumed constant at a constant reaction temperature of 200 o C. An experiment is run where the reaction proceeds at a constant temperature of T =200 o C with initial concentration of species A, 𝐶𝐶 𝐴𝐴0 = 12.2 𝑔𝑔 / 𝐿𝐿 and 𝐶𝐶 𝑝𝑝 is measured as time proceeds. Test the validity of the first order elementary rate law model for the following experimental data set. a. Fill in the missing entries in the table below: 𝑡𝑡 ( 𝑠𝑠 ) 𝐶𝐶 𝑢𝑢 𝑒𝑒𝑒𝑒𝑝𝑝𝑐𝑐 ( 𝑔𝑔 𝐿𝐿 ) 𝐶𝐶 𝑢𝑢 𝑚𝑚𝑚𝑚𝑚𝑚𝑒𝑒𝑚𝑚 ( 𝑔𝑔 𝐿𝐿 ) % relative error 0 0 10 1.36 20 2.50 30 3.81 60 6.70 120 10.73 b. Calculate the root mean squared error and coefficient of variation. RMSE=________________ 𝑦𝑦 � = __________________ CV=___________________

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Final Exam Review

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