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EXERCISE 4.36. In Example 4.20, use Eq. 4.16 to verify that the fake spheres with a = 2 and a = , have no umbilical points. (4.16) T. EXAMPLE 4.20 (Fake Spheres). We wish to construct surfaces of revolution with constant Gaussian curvature equal to 1. If K = 1, then Eq. 4.15 says that x" = -x. The following is a solution for every a > 0: (4.17) x(t) = a cos(t). The hypothesis that y is parametrizationgth, so (x')2 + (2')2 = 1, means that z is determined by x as follows: (4.18) = (t) = VI - x'(s)2 ds = 1 - a2 sin? (s) ds. (4.15) K1. Consider the simple regression model: V/i = Po+ Piri +ui, for i = 1, ..., n, with E(ur,) 7 0 and let z be a dummy instrumental variable for a, such that we can write: with E(uilz;) = 0 and E(vilzi) = 0. (c) Denote by no, the number of observations for which z =0 and by n, the number of observations for which z, = 1. Show that: (a - 2) = =(n-m). 1=1 and that: [(8 -=)(3: - 9) = 7 "(n - ni) (31 - 90) . where to and g are the sample means of y for z equal to 0 and 1 respectively. (Hint: Use the fact that n = nj + no, and that = = m). (d) Now we regress y on i to obtain an estimator of 81. From the standard formula of the slope estimator for an OLS regression and using the result in (c), show that: By1 - 90 I1 - To This estimator is called the Wald estimator.An individual has the following preferences dened oyerconsumption of a composite good c [with a per unit price p normalized to 1] and leisure h a D measured in hours: U = h "1 f 3 c" 2;" 3. Hertotal daily time constraint is H = 24 = h + l hours {where I a I] denotes her labor supply in hours] and her daily non-labor income is y ' = 8 480. Answer each of the following questions based on the above information. You have three attempts {your highest score will be recorded]. Round your FINAL numerical answers to the nearest 2nd decimal place [0.61]. {31: Assuming the hourly wage is w = S 5, calculate this individual's daily consumption of leisure, h. 02: Again, assuming the houdy wage is w = $ 5, calculate her daily consumption ofthe composite good, c. [33: Assuming the hourly wage isw = S 15, calculate this in dividual's daily consumption of leisure, h. DA: Again assuming the hourly wage is w = S 15, calculate this individual's daily consumption of the composite good c . DE: Calculate the threshold level ofthe wage, above which she would work [I :5 D} or {h a: H = 24]. D6: Is she happier when the going wage is lower {e.g., $5} or higher [e.g., 315}? Group of answer choices When the wage is higher. When the wage is lower. The Seneca Corporation manufactures lamps. It has set up the following standards per finished unit for direct materials and direct manufacturing labor, (Click the icon to view the standards. ) The number of finished units budgeted for January 2014 was 9,910, 9,900 units were actually produced (Click the icon to view actual data ) Assume that there was no beginning inventory of either direct materials or finished units. During the month, materials purchases amounted to 99, 100 lb, at a of $515.320. Input price variances are isolated upon purchase. Input efficiency variances are isolated at the time of usage. Read the requirements. Requirement 1. Compute the January 2014 price and efficiency variances of direct materials and direct manufacturing labor. Let's begin by calculating the actual input at the budgeted price. (Round your answers to the nearest whole dollar ) Actual input X Budgeted price E Cost Direct materials (purchases) Direct materials (usage) Direct manufacturing labor Next determine the formula and calculate the costs for the flexible budget Flexible budget cost Direct materials Direct manufacturing labor Now compute the price and efficiency variances for direct materials and direct manufacturing labor. Label each variance as favorable (F) or unfavorable (U). Choose from any list or enter any number in the input fields and then continue to the next question.Data Tableh Direct materials: 10 lb. at $5.00 per lb $ 50.00 Direct manufacturing labor: 0.5 hour at $30 per hour 15.00 Print Done Actual results in January 2014 were as follows: Direct materials: 97,000 lb. used Direct manufacturing labor: 4,900 hours $ 154,3502. Again, consider the general linear model Y = XB + , with & ~ Nn(0, o?/), where the first column of X consists of all ones. (a) Using facts about the mean and variance/covariance of random vectors given in lecture, show that the least squares estimate from multiple linear regression satisfies E(B) = B and Var(B) = 03(XTX)-1. (b) Let H = X(X X)-1XT be the hat matrix, Y = HY be the fitted values and e = (I - H)Y be residuals. Using properties derived in class, show that n Ex = 0. i=1 This fact is used to provide the ANOVA decomposition SSTO = SSE + SSR for multiple linear regression. (Hint: The sum above can be written as e Y. Apply properites of H.)1. Some (More) Math Review a) Let N = 3. Expand out all the terms in this expression: Cov Xi b) Now write out all the terms using the formula from last class for variance of a sum: Var( X:) = _Var(X) + > > Cov(X, X;) i-1 1=1 i-lj=1ifi Verify that (a) and (b) are giving you the same thing. Hint: Cov(X, X) = Var(X). c) Suppose that D is a roulette wheel that takes on the values {1, 2, 3} all with probability 1/3. What is the expected value of this random variable? What about the variance? d) Now suppose that M is another roulette wheel that takes on the values {1, 2,3} all with probability 1/3. Solve for the expected value of 1/M. e) Finally, suppose that D and M are independent. Solve for: E Hint: You do not need to do any new calculations here. Just remember that for independent RVs, E(XY) = E(X)E(Y). f) Does E(D/M) = E(D)/ E(M)
