0.02 0.01 O- -O y = P(d) d (mi) 100 200 300 400 i. How long did Julia's trip take? ii. Write a mathematical equation involving the function P representing the following sentence. Your answer may involve definite integrals. It takes Julia turice as long to travel the first hundred miles as it takes for her to travel the next hundred miles. iii. Let 7(d) be the time, in hours, it takes Julia to travel the first d miles on the highway. Carefully sketch a graph of T(d). iv. Explain the relationship between the functions P(d) and T(d). (c) Julia arrives, promptly buys a big mug of hot chocolate, and puts the mug down to cool. Let h(t) be the temperature, in degrees Fahrenheit, of her hot chocolate f minutes later. Use complete sentences to give practical interpretations of the equations in parts i. ili. below. i. h(0) = 160 ii. [. W'(t) dt = -34 in. . [ h(t) dt = 142 Iv. Assuming that all of the above are true, what temperature was Julia's hot chocolate 10 minutes arter she put the mug down to cool? +1. Julia recently received a GPS navigation system. (a) To test the accuracy of the GPS, she records her velocity at various times during her commute to work, then compares this information with the GPS's distance estimate. Let v(() be Julia's velocity, in miles per hour (mi/h), at time t hours (h) after she begins driving to work. Julia records several values of v(t) in the following table. t (h) 0.06 0.11 0.16 0.21 0.26 v(t) (mi/h) 60 59 56 55 50 During Julia's drive, the car is always decelerating. i. Use a left Riemann sum with four equal subintervals to estimate the distance Julia traveled between =0.06 h and t = 0.26 h. ii. Is your estimate from part (a) an overestimate or an underestimate? ifi. Julia's GPS estimates that between ? = 0.06 h and t = 0.26 h she traveled 10.8 mi, and it says that its estimate is correct to within 0.1 mi. Could the GPS be correct? Explain. (b) The following day Julia drives to the Zahir, a hip new ski resort in Michigan's Upper Peninsula. During the exhausting 400-mile drive, she tracks her pace.! In particular, she finds that, when she has traveled d miles on the highway, her pace, in hours per mile (h/mi), can be modeled by the function P(d), which is graphed below. 1/ (h/mi) 0.03 0.02 0.01 O- -0 1 = P(d) T -d (ml) 100 200 300 400 1. How long did Julia's trip take? il, Write a mathematical equation involving the function P representing the following sentence. Your answer may involve definite integrals. It takes Julia twice as long to travel the first hundred miles as it takes for her toIf A() is a strictly positive supermartingale, then zero-coupon bond prices can be modelled using the formula B(, T) = =PLA() | F,] where P is a suitably-chosen A(1) probability measure. (i) (a) Express mathematically the fact that A(t) is a strictly positive supermartingale. (b) Verify that the function A(1) =e-0.05/+0.02#(), where W() denotes standard Brownian motion, satisfies the properties in (1)(a). (c) State why the supermartingale property is required. (d) Write down the name given to this type of process. [7] (ii) By writing A() in the form A(() =e"), or otherwise, show that A(t) satisfies a stochastic differential equation of the form: dA(!) = A(D)[u(D)di +(D)dw()] State the forms of the functions #(t) and GA(!). [4] (iii) (a) Write down or derive a formula for B(t, 7) based on the process A(!) specified in (i)(b). (b) Write down expressions for the instantaneous forward rate f(t,7) and the spot rate R(t, 7) based on this model. (c) State one problem that this model of interest rates has. [4] (iv) Calculate the prices at time 5 according to the model in (ii) of the following risk- free bonds: (a) a 10-year zero-coupon bond (b) a 10-year bond that pays a coupon of 5% at the end of each year. [4] [Total 19]II. (20 Pts) Briefly answer the following questions. a. Define "Quality". How could effective quality improvement help increase productivity and reduce cost? b. Dr. Deming's point eleven is "Eliminate numerical quotas and work standards." Why should they be eliminated from the work place? C. What are chance causes and assignable causes relative to a control chart? When a point fell outside of the control limits, what would be the likely cause of that? Explain. d. When using an acceptance sampling plan, could a rejected lot have better quality (less defectives) than an accepted lot if both were from the same vendor? Why? III. (10 pts) A fishing line manufacturer claims that his new X-25 series has an average tensile strength (u) of 25 psi To verify his claim, a sample of 25 specimens was tested and yielded the following results: 32, 19, 27, 24, 28, 29, 38, 30, 25, 33, 18, 25, 37, 28, 20, 32, 23, 21, 25, 19, 23, 27, 32, 26, 30 a. Is the sample data following a normal distribution? Explain why. What is the mean and standard deviation of this sample? b. If the specification is 25 + 5 psi, what's the percentage of this fishing line not meeting the specifications? How could this percentage be lowered? IV. (10 pts) An experiment was performed to compare the burning times of chemical flares with two different formulations. The results are: Type 1 65 83 64 65 86 82 85 74 90 72 66 Type 2 64 73 57 69 67 56 76 77 89 64 59 Assuming the variances are equal. Answer the followings with an a of 0.05. a. Test the hypotheses Ho: HI " (2 VS. Hi: HI > ME Quote a p-value. b. Verify the equal variance assumption. Quote a p-value. V. (10 pts) The yield of a chemical process is related to the concentration of the reactant (3 levels) and the operating temperature (2 levels). A factorial experiment has been conducted with the results shown below. Temperature Concentration 150 F 81, 83. 79. 76 83, 85, 84, 81 89, 86. 88. 90 180"F 85, 86, 82, 80 83, 88. 85, 88 81. 78, 82, 79 Conduct an analysis of variance on this experiment including the two factors and their interaction. State appropriate hypotheses and draw your conclusions based on an a of 0.05. b. What are your recommendations regarding the concentration of the reactant and the operating temperature if high yield is desired? (Construct necessary interaction/main effect plots to support your answer) VI. (20 pts) Parts manufactured by an injection molding process are subjected to a compressive strength test. Data are given on the following 15 samples of five observations each. Sample Observations (injpsa) No. 65.9 78 2 69 8 839 78 5 7916 71.0 81.6 71.A 78. 1Question 1 (6 Marks) Robert has been saving money for a long time and finally he decides to place his saving into the bank account. Suppose from his saving Robert has deposited $12000 in Bank of the South Pacific which has the required reserve ratio of 1096, Using the information given above, calculate the following a) Money multiplier. (2 Marks) b) Required reserves. (2 Marks) c) Excess reserves, assuming the actual reserve for the bank is $2800. (2 Marks) Question (6 Marks) "Assume that the Fijian government removes all restrictions in relation to immigration which attracts a lot of people from different countries to settle in Fiji" Using the labor demand and supply curve explain what will happen to wage rate, employment and potential GDP in Fiji Question 4 (6 Marks) "The foreign exchange rate is highly affected by any change in the demand for the exported goods and services." Using foreign exchange demand and supply curve, explain the effect of decrease in world demand for domestic export on the foreign exchange market. Question 5 (6 Marks) "Open market operation refers to a central bank's buying and selling of government securities in the open market in order to expand or contract the amount of money in the banking system." Using AD/AS model, explain and show the process of an open market sales of government securities by the central bank on price level and output in the economy. Question & (6 Marks) "Monetary policy is a macroeconomic policy laid by the central bank which involves the management of money supply and interest rate to achieve the macroeconomic objectives." Using money supply and demand curve, show the effect of a contractionary monetary policy on the quantity of money and the price level