In a calorimetry experiment to determine the specific heat capacity of a metal block, the following data was recorded: Quantity Mass of the metal block 0.50 kg Mass of empty calorimeter + Stirrer 0.06 kg Mass of calorimeter + stirrer + water 0.20 kg Mass of water 0.14 kg Initial Temperature of metal block 55.5 ⁰C Initial Temperature of water and calorimeter 22 ⁰C Final Temperature of block- water system 27.4 ⁰C Take the specific heat capacity of water to be 4186 J/Kg °C. The calorimeter and stirrer are made of Copper. Specific heat capacity of Copper is 387 J/Kg °C. Ignore the mass of the stirrer. Use the data and the information given to: a) Calculate the specific heat capacity of the metal block. (Please show ALL work for full credit) b) From your result above, what is the metal block probably made up of? c) Calculate the final temperature of the block – water system if the mass of the water in the calorimeter is increased to 0.50 kg (the same as the mass of the metal block).

The following data was collected in a laboratory experiment conducted at 305.2 K.

A.) Determine the complete rate law for this simple chemical process A → B

B.) For the set of kinetic data in the problem above, if the rate doubles when the temperature increases to 415.4 K, what is the activation energy for the process?

A factorial experiment was designed to test for any significant differences in the time needed to perform English to foreign language translations with two computerized language translators. Because the type of language translated was also considered a significant factor, translations were made with both systems for three different languages: Spanish, French, and German. Use the following data for translation time in hours.

Test for any significant differences due to language translator system (Factor A), type of language (Factor B), and interaction. Use a =.05.

An experiment is performed to study the fatigue performance of a high strength alloy. The number of cycles to crack initiation is measured for twenty specimens over a range of applied pseudo-stress amplitude (PSA) levels. Use the data in the table provided to fit the following three regression models with y = Cycles and x = PSA (note the natural log transform of y for all models):

PSA (x) Cycles (y)

80, 97379

80, 340084

80, 246163

80, 239348

100, 34346

100, 23834

100, 70423

100, 51851

120, 9139

120, 9487

120, 8094

120, 17956

140, 5640

140, 3338

140, 6170

140, 5608

160, 1723

160, 3525

160, 2655

160, 1732

iii. A simple linear regression model with a logarithm transformation on PSA: lny=δ0+δ1∙ln⁡(x) .

1) Using model (iii.), find a 95% confidence interval for the mean Cycles to crack initiation at a PSA of 130

1. A paper in the Journal of the Association of Asphalt Paving Technologies (1990) described an experiment to determine the effects of air voids on percentage retained strength of asphalt. For purposes of the experiment, air voids are controlled at three levels: low (2-4%), medium (4-6%), and high (6-8%). The data are shown in the following table:

1. Write down the Null and Alternative hypotheses to test if the different levels of air voids significantly affect mean retained strength. Define all the terms used.

2. Find the calculated Ftest statistic then answer what are the critical F value and the rejection region of H0 at α=5% ?

1. c. Test the hypotheses using the test statistic / critical value method (use your answers to (b), and (c)). Clearly interpret the test result in the context of the problem.

d. What is the P value for the test? Test the hypotheses using the P value. Do you get the same answer as (d)? If your answer is different, clearly interpret it.

In an agricultural experiment, the effects of two fertilizers on the production of oranges were measured. Seventeen randomly selected plots of land were treated with fertilizer A. The average yield, in pounds, was 457 with a standard deviation of 38. Twelve randomly selected plots were treated with fertilizer B. The average yield was 394 pounds with a standard deviation of 23. Find a 99% confidence interval for the difference between the mean yields for the two fertilizers. (Round down the degrees of freedom to the nearest integer and round the final answers to two decimal places.)

3. Let the experiment be the toss of three dice in a row. Let X be the outcome of the first die. Let Y be the outcome of the 2nd die. Let Z be the outcome of the 3rd die. Let A be the event that X > Y , let B be the event that Y > Z, let C be the event that Z > X.

(a) Find P(A).

(b) Find P(B).

(c) Find P(A ∩ B).

(d) Are A and B independent?

(e) Are A, B, C pairwise independent?

(f) Find P(A ∩ B ∩ C).

(g) Are A, B, C mutually independent?

An experiment is performed to study the fatigue performance of a high strength alloy. The number of cycles to crack initiation is measured for twenty specimens over a range of applied pseudo-stress amplitude (PSA) levels. Use the data in the table provided to fit the following three regression models with y = Cycles and x = PSA (note the natural log transform of y for all models):

PSA Cycles

80        97379

80        340084

80        246163

80        239348

100      34346

100      23834

100      70423

100      51851

120      9139

120      9487

120      8094

120      17956

140      5640

140      3338

140      6170

140      5608

160      1723

160      3525

160      2655

160      1732

i. A simple linear regression model: lny=β0+β1∙x .

ii. A quadratic polynomial model: lny=γ0+γ1∙x+γ2∙x2 .

iii. A simple linear regression model with a logarithm transformation on PSA: lny=δ0+δ1∙ln⁡(x) .

1. 1. For model (i.), what hypothesis being tested by the ANOVA F-test (in terms of the coefficients of the model)? Interpret the conclusion of this test in the context of the engineering problem.
2. 2. For model (ii.), what hypothesis being tested by the ANOVA F-test (in terms of the coefficients of the model)? Interpret the conclusion of this test in the context of the engineering problem.
   3. What is the p-value for testing the significance of the quadratic term in model (ii.) (Ho: γ2=0)? Interpret the conclusion of this test in the context of the engineering problem.
3. 4. Briefly discuss the advantages and disadvantages of each of the three models.

An experiment was devised to test whether the parameter λ of a sample from the density f(y) = yeλy, y > 0 is equal to a believed value λ0 = 50. (a) Derive the most powerful test for the null hypothesis H0 : {λ = λ0} vs alternative hypothesis Ha : {λ = λa} for λa = 40. (b) Discuss whether this test is uniformly most powerful to test against a composite alternative Ha : {λ < λ0}.