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}.

A student runs an experiment with two carts on a low-friction track. As measured in the Earth reference frame, cart 1 (m = 0.48 kg ) moves from left to right at 1.0 m/s as the student walks along next to it at the same velocity. Let the +x direction be to the right.

a) What velocity v⃗ E2,i in the Earth reference frame must cart 2 (m = 0.16 kg ) have before the collision if, in the student's reference frame, cart 2 comes to rest right after the collision and cart 1 travels from right to left at 0.33 m/s?

b) What does the student measure for the momentum of the two-cart system?

c) What does a person standing in the Earth reference frame measure for the momentum of each cart before the collision?

A group of engineers perform an experiment to determine whether the fuel efficiency for a new type of engine for an oilfield drill is better than the most commonly used type of engine in existing oilfields. The engineers perform a 2–sample t–test and obtain a p-value of 0.04.

(i) Briefly explain what a p–value of 0.04 means (i.e. what does the p–value actually represent).

(ii) Explain what a Type II error would mean in the context of this problem.

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, type of language, and interaction. Use α = 0.05.

Find the value of the test statistic for language translator. (Round your answer to two decimal places.)

Find the p-value for language translator. (Round your answer to three decimal places.)

p-value =

State your conclusion about language translator.

Because the p-value ≤ α = 0.05, language translator is significant.Because the p-value ≤ α = 0.05, language translator is not significant.    Because the p-value > α = 0.05, language translator is not significant.Because the p-value > α = 0.05, language translator is significant.

Find the value of the test statistic for type of language. (Round your answer to two decimal places.)

Find the p-value for type of language. (Round your answer to three decimal places.)

p-value =

State your conclusion about type of language.

Because the p-value > α = 0.05, type of language is not significant.Because the p-value ≤ α = 0.05, type of language is not significant.    Because the p-value > α = 0.05, type of language is significant.Because the p-value ≤ α = 0.05, type of language is significant.

Find the value of the test statistic for interaction between language translator and type of language. (Round your answer to two decimal places.)

Find the p-value for interaction between language translator and type of language. (Round your answer to three decimal places.)

p-value =

State your conclusion about interaction between language translator and type of language.

Because the p-value > α = 0.05, interaction between language translator and type of language is not significant.Because the p-value ≤ α = 0.05, interaction between language translator and type of language is not significant.    Because the p-value ≤ α = 0.05, interaction between language translator and type of language is significant.Because the p-value > α = 0.05, interaction between language translator and type of language is significant.

An experiment was conducted to see the effectiveness of two antidotes to three different doses of a toxin. The antidote was given to a different sample of participants five minutes after the toxin. Twenty-five minutes later the response was measured as the concentration in the blood. What can the researchers conclude with an α of 0.05?

a) What is the appropriate test statistic?
---Select--- na one-way ANOVA within-subjects ANOVA two-way ANOVA

b) Obtain/compute the appropriate values to make a decision about H₀.
Antidote: critical value =  ; test statistic =  
Decision:  ---Select--- Reject H0 Fail to reject H0

Dose: critical value =  ; test statistic =  
Decision:  ---Select--- Reject H0 Fail to reject H0

Interaction: critical value =  ; test statistic =  
Decision:  ---Select--- Reject H0 Fail to reject H0


c) Compute the corresponding effect size(s) and indicate magnitude(s).
Antidote: η² =  ;  ---Select--- na trivial effect small effect medium effect large effect
Dose: η² =  ;  ---Select--- na trivial effect small effect medium effect large effect
Interaction: η² =  ;  ---Select--- na trivial effect small effect medium effect large effect


d) Make an interpretation based on the results.

There is an antidote difference in blood concentration.There is no antidote difference in blood concentration.     

There is a dose difference in blood concentration.There is no dose different in blood concentration.     

There is an antidote by dose interaction in blood concentration.There is no antidote by dose interaction in blood concentration.     

An experiment is planned where an automatic lab would be sent to the surface of Saturn. If there was life in Saturn, the probability that the lab detects it and correctly reports the finding is 0.5. If there never was life on Saturn, the probability that the lab will erroneously indicate the presence of life is 0.45. Suppose that a fair assessment of the probability that life was ever present on Saturn is 0.1.

(a) Find the probability that the lab says that there was life on Saturn.

(b) If the lab says that there was life on Saturn, what is the probability that the lab is correct (that is, that indeed there was life in Saturn)?