Solve, these questions as provided below.

Assignment .: 1 Q search Question 3 Automatic Joom 6 marks (a) Draw a sketch of the functions y = er and y = x* to see that there is a point of intersection near [1 mark) (b) Formulate this as a Newton's Method numerical approximation problem by first i. Formulating the function required ii. and (1 mark) Formulating the step equation giving the next value from the previous (2 marks) (c) Apply Newton's Method 3 times to approximate where the graphs intersect to 3 decimal places (2 marks) Question 4 6 marks Consider a function that can be written as a polynomial such as f(x) = >lod, x* = do tax + a, x + + a, r" where the [a, : k = 0, 1, .. n] are called the coefficients of each term. (a) By computing the values of f(0), /'(0), /"(0) etc, find an expression for a, in terms of the differentials of f evaluated at x = 0. (2 marks) (b) Consider / (x) = (1 + x)". Compute each of the values of /(0), /'(0). /"(0) etc (2 marks) (c) Hence verify the Binomial Theorem expansion : (1 + )" =)()x* where n! = n x (n - 1) x (n - 2) x_.3 x 2x1 and 0! = 1 [2 marks] Question 5 8 marks A manufacturing company produces the quantity q of a product that depends on W workers and a capital amount K as given by the equation q =6 (W): (K). Labour costs are $10 per worker and capital costs are $20 per unit of capital and the total cost is $3000. (a) Formulate the Lagrange function for this problem. (1 mark) (b) Find the partial derivatives of the Lagrange function. (1 mark] (c) Solve the system of equations to find the optimal solution. (2 marks] [d) Demonstrate the following economic principle for the optimal solution found in [c) that The marginal productivity of labour () : The marginal productivity of capital () = Cost per unit of labour : Cost per unit of capital [2 marks] (e) Recompute the optimal values for W and & when the total cost is increased by $1 and check that this [2 marks) allows for the production of an extra A units where A is the Lagrangian multiplier.Q4. (a) Survival times are available for four insureds, two from risk class A and two from risk class B. The two from risk class A passed away at times t = 1 and t = 9. The two from risk class B died at times t = 2 and t = 4. Using the Empirical Credibility Model I, estimate the credibility factor Z. [Marks 8] (b) An investigative journalist comes across information about two policy groups administered by an insurance company. The data however has missing values for some years in total losses and numbers in policy groups as shown in the following table. The missing values are represented by the symbol * Policy group Year 1 Year 2 Year 3 Year 4 Total Losses 750 600 Number in group 3 2 Total Losses 975 1200 900 Number in group 5 6 4 5 Using the Empirical Credibility Model II, determine the premium for each policyholder in Year 4. [Marks 8]Consider the AD-AS (Aggregate Demand-Aggregate Supply) model in a closed economy. The economy is characterized by the following equations: . AD= C+/+G where AD is aggregate demand (or aggregate expenditures), C is aggregate consumption, / is aggregate investment, G is government expenditures. . C=C +mpc. YD and YD=Y-T where C is autonomous consumption expenditures, mpc is the marginal propensity to consume, Yo is the disposable income, Y is aggregate output (or aggregate income), T is the taxes. . I=1-d.r where I is autonomous investment, d is a parameter that reflects the responsiveness of investment to the real interest rate, " is the real interest rate. G=gY and T=tY where g and f (tax rate) are parameters between 0 and 1. = F+1.n where f is the autonomous component of the real interest rate, 1 is a parameter that reflects the responsiveness of the real interest rate to the inflation rate. . I = n ty(Y - YP) + p where i is the expected inflation rate, y is a parameter that reflects the sensitivity of inflation to the output gap, YP is the potential output, p is the inflation shock. Assume that initially the economy is at the equilibrium with the level of aggregate output, the inflation rate and the interest rate being at YP, to and ro, respectively a. Show the initial equilibrium using SRAS (short-run aggregate supply), LRAS (long-run aggregate supply) and AD (aggregate demand) curves. Name the axes. Show your work. (6 points) b. Using the information above derive the IS curve equation by isolating r. Draw the curve of the equation, show one of the intercepts. Name the axes. Show your work. (10 points) c. Using the information above derive the AD curve equation by isolating Y. Draw the curve of the equation, show one of the intercepts. Name the axes. Show your work. (10 points) d. Assume that the oil prices have temporarily increased. Which parameter reflects this change in the oil prices? What is the impact of this change on the equilibrium values of Yand *? Explain it using a graphic and sentences. (10 points) e. Assume that the central bank decided to increase F. What type of a policy is this? What is the impact of this policy on the equilibrium values of Yand *? Firstly, draw the initial equilibrium (as in part c) and then show the shifts of the relevant curves in the short and in the long run and show the final long run equilibrium on the same graph. Name the axes. Use sentences. (10 points)MAT 124 Data Analysis if! - Spring 21:21 In this assignment, you will practice conducting a hypothesis test using real data. You will test if there is evidence that the temperature in a month and year of your choosing was different than the averages. Below are the average monthly low and high temperatures (in degrees Fahrenheit) ln New Jersey that you will use for the null hypothesis. [Em- AveraeLow AveraeHIh 49" En! Tl' Tr'El' 34 we. 82* 5e -tem her October 64\" I'- _ m _ December To Do: 1. Pick a month and year to compare to the average. Decide if you will compare low or high temperatures. 2. Research and record the daily low or high temperature for your chosen month. State the website where you found your data. 3. Find the mean iround to 1 decimal place} and standard deviation {round to .2 decimal placesi. Pick a significance level [or] for your hypothesis tests. Conduct a two-sided hypothesis test. Conduct the appropriate one-sld ed hypothesis test. Write up each hypothesis test, including an appropriate conclusion and interpretation. You will notice that the t-value for each hypothesis test is the same. but the p value is different. Explain how the t- and p-values are related to each other and why the p-vaiue changes. (Note this Is not included In the example.) I Hints: o It may be helpful to explain using a graph ofthe normal distribution. Since n :- 23. the t~dlstrlbution will be close enough to the normal distribution that you can use the normal distribution for your explanation. o There are graphs in the Chapter El module lectures in Canvas. PHF'E"