Problem 1 (3 + 3 + 3 = 9) Suppose you draw two cards from a deck of 52 cards without replacement. 1) What’s the probability that both of the cards are hearts? 2) What’s the probability that exactly one of the cards are hearts? 3) What’s the probability that none of the cards are hearts?

Problem 2 (4) A factory produces 100 unit of a certain product and 5 of them are defective. If 3 units are picked at random then what is the probability that none of them are defective?

Problem 3 (3+4=7) There are 3 bags each containing 100 marbles. Bag 1 has 75 red and 25 blue marbles. Bag 2 has 60 red and 40 blue marbles. Bag 3 has 45 red and 55 blue marbles. Now a bag is chosen at random and a marble is also picked at random. 1) What is the probability that the marble is blue? 2) What happens when the first bag is chosen with probability 0.5 and other bags with equal probability each?

Probem 4 (3+3+4=10) Before each class, I either drink a cup of coffee, a cup of tea, or a cup of water. The probability of coffee is 0.7, the probability of tea is 0.2, and the probability of water is 0.1. If I drink coffee, the probability that the lecture ends early is 0.3. If I drink tea, the probability that the lecture ends early is 0.2. If I drink water, the lecture never ends early. 1) What’s the probability that I drink tea and finish the lecture early? 2) What’s the probability that I finish the lecture early? 3) Given the lecture finishes early, what’s the probability I drank coffee?

Problem 5 (4+4+4=12) We roll two fair 6-sided dice. Each one of the 36 possible outcomes is assumed to be equally likely. 1) Find the probability that doubles were rolled. 2) Given that the roll resulted in a sum of 4 or less, find the conditional probability that doubles were rolled. 3) Given that the two dice land on different numbers, find the conditional probability that at least one die is a 1. Problem 6 (8) For any events A, B, and C, prove the following equality: P(B|A) P(C|A) = P(B|A ∩ C) P(C|A ∩ B)

The paint used to make lines on roads must reflect enough light to be clearly visible at night. Let μ denote the true average reflectometer reading for a new type of paint under consideration. A test of H₀: μ = 20 versus H_a: μ > 20 will be based on a random sample of size n from a normal population distribution. What conclusion is appropriate in each of the following situations? (Round your P-values to three decimal places.)

(a)    n = 15, t = 3.3, α = 0.05
P-value =

State the conclusion in the problem context.

Reject the null hypothesis. There is not sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20.

Reject the null hypothesis. There is sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20.  

  Do not reject the null hypothesis. There is sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20.

Do not reject the null hypothesis. There is not sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20.

(b)    n = 9, t = 1.7, α = 0.01
P-value =

State the conclusion in the problem context.

Reject the null hypothesis. There is not sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20.

Do not reject the null hypothesis. There is sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20.    

Do not reject the null hypothesis. There is not sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20.

Reject the null hypothesis. There is sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20.

(c)    n = 29,

t = −0.3

P-value =

State the conclusion in the problem context.

Reject the null hypothesis. There is sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20.

Reject the null hypothesis. There is not sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20.  

Do not reject the null hypothesis. There is not sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20.

Do not reject the null hypothesis. There is sufficient evidence to conclude that the new paint has a reflectometer reading higher than 20.

A wastewater treatment plant, which serves a population of 300,000 people, receives an average daily volume of 24 million gallons per day (MGD) at an average influent 5-day biochemical oxygen demand concentration of 200 mgBOD5/L and an average influent total suspended solids concentration of 220 mgTSS/L. The plant operates a primary sedimentation process that remove 65% of the incoming TSS and 35% of the incoming BOD5%. This process is followed by a secondary treatment process before discharge to the local waters.

9. What is plant average BOD5 loading? (answer will be given in the unit of !#. /01!") ,"-

10. Knowing the primary treatment BOD5 removal efficiency and the plant combined primary and secondary BOD5 removal efficiency. How much BOD5 is removed during secondary

    treatment every day? (answer will be given in !#. /01!") ,"-

11. The biomass is quantified in terms of volatile suspended solids concentration (VSS) . For example, if the content of a biological reactor has a VSS concentration of 2000 mgVSS/L, it is estimated that there is 2,000 mg of biomass per Liter of reactor.
    With that said, what will be the expected secondary treatment biomass production if the

    net biomass yield is 0.3 lb VSS/lb of BOD5 removed)? (answer will be given in !#. 633") ,"-

The partial pressure of oxygen in the lung alveoli is a bit lower than in ambient air, being about 100 mm of mercury, or 0.13 Atm (it is lower than the partial pressure in air mainly because oxygen is continually taken up by the alveolar capillaries and carbon dioxide is continually released into the alveoli).  In cell-free blood plasma (or a saline solution formulated to match key characteristics of blood plasma), which lacks red blood cells and therefore lacks hemoglobin, the concentration of oxygen will equilibrate at 37° C at about 0.3 ml O₂/100 ml plasma. For whole blood (with hemoglobin), however, the O₂ concentration is around 20 ml O₂/100 ml whole blood.

By what factor does the presence of hemoglobin increase the oxygen content of blood?

B.  Given the above, imagine that you are an emergency room physician treating a patient who lost a quarter of his blood in an accident. A paramedic replaced this lost blood with saline solution to keep his blood pressure up. The saline solution contains no hemoglobin since it contains no red blood cells. The patient is short of breath and oxygen levels in his blood are dangerously low. If for some reason you must choose between administering pure (100%) oxygen or giving a transfusion of whole blood to restore the red blood cell count, which would you expect to be more helpful? Address this decision by answering the questions below. Show your work and be as quantitatively explicit as possible.

1. Under normal conditions, what percentage of blood oxygen content is accounted for by oxygen dissolved in the plasma and what percentage is bound to hemoglobin? By how much do these percentages change after the patient has a quarter of his blood volume replaced with saline solution?
2. a) By what percentage could total blood oxygen content be increased by delivering pure (100%) oxygen instead of allowing the patient to simply breathe ambient air? Assume that Henry’s Law applies.

b) One obviously can’t deliver oxygen at a concentration higher than 100%, but how else might the partial pressure of the oxygen being delivered be modified to increase the amount diffusing into the blood?

3. By what percentage could blood oxygen content be increased by transfusing whole blood to restore the red blood cell count? Relative to delivering pure oxygen, is this a more promising or less promising approach for restoring normal blood oxygen levels? Explain.

A company assembles and sells skateboards. One popular model is the "ICE". The final assembly plan for April to September, which also represents 50% of a full year’s demand:

The company is using MRP. The forecast for Skateboard ICE for the next coming six weeks:

The company are buying all components from different suppliers. They are only making the wheel assembly and the assembly of the final skateboard, see also the diagram above.

The company has an ordering cost of $150/order and the inventory carrying cost is estimated to 10%.

3a. If we use moving average with n=5, what is the forecast for Skateboard ICE for October?

3b. If we use exponential smoothing with α=0.3, what is the forecast for Skateboard ICE for October?

3c. Suppose it's now week 35. In what week should production of wheel assemblies start?

3d. In relation to question 3c above, what quantity of wheel assemblies will be needed?

3e. The company have had some problems with the supplier of the truck parts and therefore want to review the setup. But first they want to check how many truck parts they should order each time by calculating the Economic Order Quantity (EOQ).

What quantity of truck parts should the company order?

3f. The supplier of the truck parts replies to the company that if they order in lots of 10 000 each time, they will get a 5 % discount.

What is the total cost for the truck parts if the company order 10 000 each time?

An assistant in the district sales office of a national cosmetics firm obtained data on advertising expenditures and sales last year in the district’s 44 territories.

X1: expenditures for point-of-sale displays in beauty salons and department stores (X$1000).

X2: expenditures for local media advertising.

X3: expenditures for prorated share of national media advertising.

Y: Sales (X$1000).

1. Test the regression relation between sales and the three predictor variables. State the hypotheses, test statistic and degrees of freedom, the p-value, the conclusion in words.

2. Determine whether the linear regression model is appropriate by using the “usual” plots (scatterplot, residual plots, histogram/QQ plot). Explain in detail whether or not each assumption appears to be substantially violated.

An assistant in the district sales office of a national cosmetics firm obtained data on advertising expenditures and sales last year in the district’s 44 territories. Data is consmetics.csv. Use R. I don't want answers in Excel or SAS :)

X1: expenditures for point-of-sale displays in beauty salons and department stores (X$1000).

X2: expenditures for local media advertising.

X3: expenditures for prorated share of national media advertising.

Y: Sales (X$1000).

6. (4) Are there any influential points?

7. Is there a serious multicollinearity problem?

(3) Include an appropriate scatterplot and correlation values between the explanatory variables.

(3) Judge by VIF, do you think there is a problem with multicollinearity? (Hint: VIP or tolerance)

(3) Compare your answers in parts i and ii. Are your conclusions the same or different? Please explain your answer.

Data:

Flexible Budgeting and Variance Analysis

I Love My Chocolate Company makes dark chocolate and light chocolate. Both products require cocoa and sugar. The following planning information has been made available:

I Love My Chocolate Company does not expect there to be any beginning or ending inventories of cocoa or sugar. At the end of the budget year, I Love My Chocolate Company had the following actual results:

Required:

1. Prepare the following variance analyses for both chocolates and the total, based on the actual results and production levels at the end of the budget year:

     a. Direct materials price variance, direct materials quantity variance, and total variance.

     b. Direct labor rate variance, direct labor time variance, and total variance.

Enter a favorable variance as a negative number using a minus sign and an unfavorable variance as a positive number.

2. The variance analyses should be based on the standard  amounts at actual  volumes. The budget must flex with the volume changes. If the actual  volume is different from the planned volume, as it was in this case, then the budget used for performance evaluation should reflect the change in direct materials and direct labor that will be required for the actual  production. In this way, spending from volume changes can be separated from efficiency and price variances.

When an airbag explodes, there are 3 different types of reactions that occur. Sodium azide produces nitrogen gas but there is a bi-product of Na. Na is very reactive and must be neutralized. For this, potassium nitrate is used. This creates two further compounds, sodium oxide and potassium oxide, which must be neutralized by silicon dioxide.

Chemical reactions:

1. Sodium Azide is ignited. Nitrogen gas fills nylon bag at 150-250 miles/hr

NaN₃ ? N₂ + Na

2. Reaction with potassium nitrate (1st stage to eliminating dangerous by-products)

Na + KNO₃ ? N₂ + Na₂O + K₂O

3. Reaction with sodium and potassium oxide to form silicate glass (2nd stage to eliminating dangerous by-products)

K₂O + SiO₂ ? K₄SiO₄ Na₂O + SiO₂ ? Na₄SiO₄

A typical 60L airbag requires 5.82 moles of nitrogen gas to fill it up. A manufacturer puts 65.0 g of SiO₂ in an airbag. Using stoichiometry, we are going to find out how many grams of SiO₂ is required to completely neutralize the dangerous by-products of the airbag reaction & conclude whether 65.0 g is enough.

PART A:

1. Use stoichiometry to calculate the number of moles of sodium produced by the first reaction if 378.3g of NaN₃ is used. SHOW ALL YOUR WORK & BE NEAT!! Use significant figures where appropriate.

NaN₃? N₂+Na

PART B:

2. Sodium is very reactive and must be neutralized. Using the number of moles of Na produced from the first reaction, calculate using stoichiometry. SHOW ALL YOUR WORK & BE NEAT!! Use significant figures where appropriate.

Na + KNO3 ? N2 + Na2O + K2O

2a) how many moles of Na2O are created?

2b) how many moles of K2O are created?

PART 3 ; SHOW ALL YOUR WORK AND BE NEAT.

The products Na₂O + K₂O are also dangerous, and must further be neutralized by SiO₂ to produce K₄SiO₄ and Na₄SiO₄

3a) What mass of SiO₂ would be required in order to fully react with all of the of K₂O from question (2)?

K₂O + SiO2 ? K₄SiO₄

3b) What mass of SiO₂ would be required in order to fully react with all of the of Na₂O from question (2)

Na2O + SiO2 ? Na₄SiO₄

4. How much SiO₂ is needed in total? Was 65 g of SiO₂ enough?

Ramp metering is a traffic engineering idea that requires cars entering a freeway to stop for a certain period of time before joining the traffic flow. The theory is that ramp metering controls the number of cars on the freeway and the number of cars accessing the freeway, resulting in a freer flow of cars, which ultimately results in faster travel times. To test whether ramp metering is effective in reducing travel times, engineers conducted an experiment in which a section of freeway had ramp meters installed on the on-ramps. The response variable for the study was speed of the vehicles. A random sample of 15 cars on the highway for a Monday at 6 p.m. with the ramp meters on and a second random sample of 15 cars on a different Monday at 6 p.m. with the meters off resulted in the following speeds (in miles per hour).

Does there appear to be a difference in the speeds?

A.Yes, the Meters Off data appear to have higher speeds.

B.Yes, the Meters On data appear to have higher speeds.

C.No, the box plots do not show any difference in speeds.

Are there any outliers?

A.Yes, there appears to be a high outlier in the Meters On data.

B.No, there does not appear to be any outliers.

C.Yes, there appears to be a low outlier in the Meters On data.

D.Yes, there appears to be a high outlier in the Meters Off data.

Are the ramp meters effective in maintaining a higher speed on the freeway? Use the alphaαequals=0.01 0.01 level of significance. State the null and alternative hypotheses. Choose the correct answer below.

Determine the P-value for this test.

Choose the correct conclusion

A researcher wanted to determine if carpeted rooms contain more bacteria than uncarpeted rooms. The table shows the results for the number of bacteria per cubic foot for both types of rooms.

State the null and alternative hypotheses. Let population 1 be carpeted rooms and population 2 be uncarpeted rooms.

Determine the P-value for this hypothesis test.(round to 3 decimals)

State the appropriate conclusion. Choose the correct answer below.

The data is

Carpeted: 15.3,12.9,10.2,6.9,15.6,12.7,10.6,14.6

Uncarpeted;8.7,10,11.2,10.7,14,6.9,6.4,11.1