Problem 3a: (6pts) As shown below, a two-dimensional vector can be defined by its x and y coordinates in an x-y Cartesian coordinate system. The vector can also be defined by its x ‘ and y ‘ coordinates in an x ‘− y’ Cartesian coordinate system that is rotated by a positive angle θ with respect to the x-y Cartesian coordinate system The relation between the two sets of coordinates, (x, y) and ( x ‘ , y ‘), is defined by the following transformation: A x y ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = x ‘ y ‘ ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ where the transformation matrix is given by: A = cosθ sinθ −sinθ cosθ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ (i) Determine the inverse of the matrix A. (4 pts) (ii) Show that the inverse determined in Part (a) is the transformation matrix corresponding to rotation by a negative angle (i.e., rotation by –θ). (2 pts) Hint: cos(θ) = cos (−θ) and sin (θ) = −sin (−θ)

Problem 3a: (6pts) As shown below, a two-dimensional vector can be defined by its x and y coordinates in an x-y Cartesian coordinate system. The vector can also be defined by its x ‘ and y ‘ coordinates in an x ‘− y’ Cartesian coordinate system that is rotated by a positive angle θ with respect to the x-y Cartesian coordinate system The relation between the two sets of coordinates, (x, y) and ( x ‘ , y ‘), is defined by the following transformation: A x y ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = x ‘ y ‘ ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ where the transformation matrix is given by: A = cosθ sinθ −sinθ cosθ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ (i) Determine the inverse of the matrix A. (4 pts) (ii) Show that the inverse determined in Part (a) is the transformation matrix corresponding to rotation by a negative angle (i.e., rotation by –θ). (2 pts) Hint: cos(θ) = cos (−θ) and sin (θ) = −sin (−θ)

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When the sheep Dolly was successfully cloned, it was produced by growing an in vitro fertilized egg where the normal egg nucleus had been removed and replaced by a nucleus from an adult. Since this nucleus is from an old mature animal, we would expect it to ______. Interestingly, tests show that this did not happen, a fact that currently puzzles researchers. Select one: have additional Barr bodies be mutated have shorter telomeres have longer telomeres express transcription and translation more rapidly

When the sheep Dolly was successfully cloned, it was produced by growing an in vitro fertilized egg where the normal egg nucleus had been removed and replaced by a nucleus from an adult. Since this nucleus is from an old mature animal, we would expect it to ______. Interestingly, tests show that this did not happen, a fact that currently puzzles researchers. Select one: have additional Barr bodies be mutated have shorter telomeres have longer telomeres express transcription and translation more rapidly

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Problem 5: Physical Fitness versus Weight. You may have noticed from your analysis in Problem 4 that height does not explain 100% of the variation that we have observed in students’ heights. Is it possible that the amount of time students devote to physical fitness each week may help us to better understand their weights? a. Question 12 of the survey asked students, “About how much time per week (on average) do you devote to physical fitness?” We have named this variable FITNESS. Create a suitable graph to display the distribution of FITNESS and insert it here. b. What is the mode of this distribution? (Please underline one option.) Between 0 & 2 hours Between 2 & 5 hours Between 5 & 9 hours Between 9 & 15 hours Over 15 hours c. Create side-by-side boxplots to display students’ weights for the different levels of FITNESS. (Go to Graph > Boxplot > One Y with Groups > OK. Select WEIGHT for the “Graph variables” slot and FITNESS for the “Categorical variables for grouping” slot.) Insert your graph here. d. Use Minitab to calculate the basic statistics of WEIGHT for each level of FITNESS. Copy and paste the output here. e. With regard to FITNESS levels, which group of students has the lowest mean weight? (Please underline one option.) Between 0 & 2 hours Between 2 & 5 hours Between 5 & 9 hours Between 9 & 15 hours Over 15 hours f. Discuss the results: Describe the distributions of WEIGHT for the different levels of FITNESS as well as draw comparisons (i.e., What do they have in common?) and contrasts (i.e., How are they different?) between these distributions. Are there any surprises in the results? Explain why you think so, or why not. Problem 6 (Even): If your E number ends in an even number (0, 2, 4, 6, or 8) then do this question. (Omit this page/problem if your E# ends with an odd number.) Gender and Nuclear Safety. Question 5 in the survey asked students “How safe would you feel if a nuclear energy plant were built near where you live?” (Students could choose one of these options: Extremely safe, Very Safe, Moderately safe, Slightly safe, or Not at all safe.) Is there a relationship between gender and students’ opinions about nuclear safety? a. Create an appropriate graph to display the relationship between GENDER and NUCLEAR SAFETY. You don’t want to display information for students that didn’t answer both of these questions on the survey, so click on Data Options > Group Options and remove the checks in the boxes beside “Include missing as a group” and “Include empty cells.” Insert your graph here. b. Create an appropriate two-way table to summarize the data. Click on Options > Display missing values for… and put a dot in the circle beside “No variables.” Insert your table here. c. SUPPOSE WE SELECT ONE STUDENT AT RANDOM: (Calculate the following probabilities and show your work.) i. What is the probability that this student is a female and feels “very safe”? P = ii. What is the probability that this student is either a male or that he/she feels “very safe”? P = iii. What is the probability that this student feels “not at all safe” given that the student selected is a female? P = iv. What is the probability that this student is a male given that the student selected feels “not at all safe”? P = d. Do you think there may be an association between GENDER and NUCLEAR SAFETY? Why or why not? Explain your reasoning based on what you see in your graph.

Problem 5: Physical Fitness versus Weight. You may have noticed from your analysis in Problem 4 that height does not explain 100% of the variation that we have observed in students’ heights. Is it possible that the amount of time students devote to physical fitness each week may help us to better understand their weights? a. Question 12 of the survey asked students, “About how much time per week (on average) do you devote to physical fitness?” We have named this variable FITNESS. Create a suitable graph to display the distribution of FITNESS and insert it here. b. What is the mode of this distribution? (Please underline one option.) Between 0 & 2 hours Between 2 & 5 hours Between 5 & 9 hours Between 9 & 15 hours Over 15 hours c. Create side-by-side boxplots to display students’ weights for the different levels of FITNESS. (Go to Graph > Boxplot > One Y with Groups > OK. Select WEIGHT for the “Graph variables” slot and FITNESS for the “Categorical variables for grouping” slot.) Insert your graph here. d. Use Minitab to calculate the basic statistics of WEIGHT for each level of FITNESS. Copy and paste the output here. e. With regard to FITNESS levels, which group of students has the lowest mean weight? (Please underline one option.) Between 0 & 2 hours Between 2 & 5 hours Between 5 & 9 hours Between 9 & 15 hours Over 15 hours f. Discuss the results: Describe the distributions of WEIGHT for the different levels of FITNESS as well as draw comparisons (i.e., What do they have in common?) and contrasts (i.e., How are they different?) between these distributions. Are there any surprises in the results? Explain why you think so, or why not. Problem 6 (Even): If your E number ends in an even number (0, 2, 4, 6, or 8) then do this question. (Omit this page/problem if your E# ends with an odd number.) Gender and Nuclear Safety. Question 5 in the survey asked students “How safe would you feel if a nuclear energy plant were built near where you live?” (Students could choose one of these options: Extremely safe, Very Safe, Moderately safe, Slightly safe, or Not at all safe.) Is there a relationship between gender and students’ opinions about nuclear safety? a. Create an appropriate graph to display the relationship between GENDER and NUCLEAR SAFETY. You don’t want to display information for students that didn’t answer both of these questions on the survey, so click on Data Options > Group Options and remove the checks in the boxes beside “Include missing as a group” and “Include empty cells.” Insert your graph here. b. Create an appropriate two-way table to summarize the data. Click on Options > Display missing values for… and put a dot in the circle beside “No variables.” Insert your table here. c. SUPPOSE WE SELECT ONE STUDENT AT RANDOM: (Calculate the following probabilities and show your work.) i. What is the probability that this student is a female and feels “very safe”? P = ii. What is the probability that this student is either a male or that he/she feels “very safe”? P = iii. What is the probability that this student feels “not at all safe” given that the student selected is a female? P = iv. What is the probability that this student is a male given that the student selected feels “not at all safe”? P = d. Do you think there may be an association between GENDER and NUCLEAR SAFETY? Why or why not? Explain your reasoning based on what you see in your graph.

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Learning Goals: Students will be able to determine the gravitational acceleration of “Planet X” 1. Research to find equations that would help you find g using a pendulum. Design an experiment and test your design using Moon and Jupiter. Write your procedure in a paragraph that another student could use to verify your results. Show your data, graphs, and calculations that support your strategy. 2. Use your procedure to find g on Planet X. Show your data, graphs, and calculations that support your conclusion. 3. Give your conclusion and write an error analysis.

Learning Goals: Students will be able to determine the gravitational acceleration of “Planet X” 1. Research to find equations that would help you find g using a pendulum. Design an experiment and test your design using Moon and Jupiter. Write your procedure in a paragraph that another student could use to verify your results. Show your data, graphs, and calculations that support your strategy. 2. Use your procedure to find g on Planet X. Show your data, graphs, and calculations that support your conclusion. 3. Give your conclusion and write an error analysis.

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University of California, Los Angeles Department of Statistics Statistics 100C Instructor: Nicolas Christou Homework 4 Exercise 1 Consider the following simple regression model yi = 0 + 1xi + i, for which E(i) = 0, E(ij) = 0 for i 6= j, and var(i) = 2. The normal equations discussed earlier in class are: n^ 0 + ^ 1 Xn i=1 xi = Xn i=1 yi ^ 0 Xn i=1 xi + ^ 1 Xn i=1 x2i = Xn i=1 xiyi In matrix form this system of two equations with two unknowns can be expressed as follows:  n Pn i=1 P xi n i=1 xi Pn i=1 x2i  ^ 0 ^ 1  =  Pn i=1 P yi n i=1 xiyi  a. Use matrix algebra to nd the solution for the vector ^ = ( ^ 0; ^ 1)0. b. Use matrix algebra to nd the variance covariance matrix of the vector ^ , i.e.  var( ^ 0) cov( ^ 0; 1) cov( ^ 1; 1) var( ^ 1)  : Exercise 2 Consider the following simple regression model for which i  N(0; ). y1 = 0 + 0:5 1 + 1 y2 = 0 ? 1 + 2 y3 = 0 + 0:5 1 + 3 a. Write the above model in matrix form. b. Find the least squares estimates using vectors and matrices. c. Find the variance-covariance matrix of ^ . d. Find the hat matrix. Verify that the sum of the diagonal elements of the hat matrix is equal to 2 ( Pn i=1 hii = k + 1). e. Generate your own data with n = 3 based on this model and verify that the estimates of 0 and 1 are those given by part (b). Exercise 3 Suppose that you need to t the multiple regression model yi = 0 + 1x1i + 2x2i + i, where E(i) = 0, E(ij) = 0 for i 6= j, and var(i) = 2, to the following data: y x1 x2 -43.6 27 34 3.3 33 30 -12.4 27 33 7.6 24 11 11.4 31 16 5.9 40 30 -4.5 15 17 22.7 26 12 -14.4 22 21 -28.3 23 27 It turns out that (X0X)?1 = 0 @ 1:97015 ?0:05623 ?0:01572 ?0:05623 0:00289 ?0:00091 ?0:01572 ?0:00091 0:00174 1 A and X0Y = 0 @ ?52:3 ?1076:3 ?2220:2 1 A a. Find the least squares estimator of = ( 0; 1; 2)0. b. Find the variance-covariance matrix of the previous estimator. c. Compute the estimate s2e of 2. d. Using your answers to parts (b) and (c) nd the variances of ^ 0; ^ 1, and ^ 2. e. Find the tted value ^y1 a nd its variance. f. What is the variance of the rst residual (var(ei))? Exercise 4 Show that the residuals are orthogonal to the matrix X as well as to the tted values ^Y . This is true for simple or multiple regression models. a. e0X = 0. b. e0^Y = 0. c. Use part (a) to show the already known result that Pn i=1 ei = 0.

University of California, Los Angeles Department of Statistics Statistics 100C Instructor: Nicolas Christou Homework 4 Exercise 1 Consider the following simple regression model yi = 0 + 1xi + i, for which E(i) = 0, E(ij) = 0 for i 6= j, and var(i) = 2. The normal equations discussed earlier in class are: n^ 0 + ^ 1 Xn i=1 xi = Xn i=1 yi ^ 0 Xn i=1 xi + ^ 1 Xn i=1 x2i = Xn i=1 xiyi In matrix form this system of two equations with two unknowns can be expressed as follows:  n Pn i=1 P xi n i=1 xi Pn i=1 x2i  ^ 0 ^ 1  =  Pn i=1 P yi n i=1 xiyi  a. Use matrix algebra to nd the solution for the vector ^ = ( ^ 0; ^ 1)0. b. Use matrix algebra to nd the variance covariance matrix of the vector ^ , i.e.  var( ^ 0) cov( ^ 0; 1) cov( ^ 1; 1) var( ^ 1)  : Exercise 2 Consider the following simple regression model for which i  N(0; ). y1 = 0 + 0:5 1 + 1 y2 = 0 ? 1 + 2 y3 = 0 + 0:5 1 + 3 a. Write the above model in matrix form. b. Find the least squares estimates using vectors and matrices. c. Find the variance-covariance matrix of ^ . d. Find the hat matrix. Verify that the sum of the diagonal elements of the hat matrix is equal to 2 ( Pn i=1 hii = k + 1). e. Generate your own data with n = 3 based on this model and verify that the estimates of 0 and 1 are those given by part (b). Exercise 3 Suppose that you need to t the multiple regression model yi = 0 + 1x1i + 2x2i + i, where E(i) = 0, E(ij) = 0 for i 6= j, and var(i) = 2, to the following data: y x1 x2 -43.6 27 34 3.3 33 30 -12.4 27 33 7.6 24 11 11.4 31 16 5.9 40 30 -4.5 15 17 22.7 26 12 -14.4 22 21 -28.3 23 27 It turns out that (X0X)?1 = 0 @ 1:97015 ?0:05623 ?0:01572 ?0:05623 0:00289 ?0:00091 ?0:01572 ?0:00091 0:00174 1 A and X0Y = 0 @ ?52:3 ?1076:3 ?2220:2 1 A a. Find the least squares estimator of = ( 0; 1; 2)0. b. Find the variance-covariance matrix of the previous estimator. c. Compute the estimate s2e of 2. d. Using your answers to parts (b) and (c) nd the variances of ^ 0; ^ 1, and ^ 2. e. Find the tted value ^y1 a nd its variance. f. What is the variance of the rst residual (var(ei))? Exercise 4 Show that the residuals are orthogonal to the matrix X as well as to the tted values ^Y . This is true for simple or multiple regression models. a. e0X = 0. b. e0^Y = 0. c. Use part (a) to show the already known result that Pn i=1 ei = 0.

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QUESTION 1 1. Convert 206 degrees 9 minutes and 15 seconds to decimal degrees. Show your answers to only 6 decimal places. Do not give units. 1 points QUESTION 2 1. COMPUTE the sin of 68 degrees. Give the answer to 6 decimal places. 1 points QUESTION 3 1. What is the sine of 83 degrees and 37 minutes? Give your answer to 6 decimal places. Pay attention to rounding. 1 points QUESTION 4 1. This is a right triangle problem. Angle A is 90 degrees. Draw the triangle and label it as we did in lecture. If angle B is 64 degrees 15 minutes and side c is 332.98 feet, what is the distance in feet of side b? Give your answer to two decimal places. Do not provide units. Those are in feet – right? 1 points QUESTION 5 1. This is a right triangle problem with angle A being the 90 degree angle. It should look like the one from lecture. If angle B is 31 degrees 10 minutes and side c is 312.86 feet, what is the distance to two decimal places of side a? Give your answer to two decimal places. Do not provide units – those are in feet. 1 points QUESTION 6 Ad by Browse Safe | Close 1. It is desired to determine the height of a flagpole. Assuming that the ground is level, an instrument is set up 216.46 feet from the flagpole with its telescope centered 4.92 feet above the ground. The telescope is sighted horizontally to a point 4.92 feet from the bottom of the flagpole and then the angle at the instrument looking to the top of the pole is measured. That angle is 25 degrees 34 minutes. How tall is the flagpole from its base? Give your answer to two decimal places with NO units. 1 points QUESTION 7 1. You are hiking in the mountains. For every 100.00 feet you would be walking horizontally, you have increased your elevation by 4.6 feet. At what grade are you climbing? Give your answer to three decimal places. Hint: Your units will be in ft/ft. 1 points QUESTION 8 1. A grade of 0.4 percent is being considered for a mountain roadway. The elevation at the initial point is 2,054.23 feet and a horizontal distance of 5,758.79 needs to be covered. What is the elevation at the end of the grade? 1 points QUESTION 9 1. A slope distance was measured between two points (A and T) and determined to be 3,307.97 feet. At point A the elevation is 872.17 feet and at point T the elevation is 884.21 feet. What is the horizontal distance between A and T? 1 points

QUESTION 1 1. Convert 206 degrees 9 minutes and 15 seconds to decimal degrees. Show your answers to only 6 decimal places. Do not give units. 1 points QUESTION 2 1. COMPUTE the sin of 68 degrees. Give the answer to 6 decimal places. 1 points QUESTION 3 1. What is the sine of 83 degrees and 37 minutes? Give your answer to 6 decimal places. Pay attention to rounding. 1 points QUESTION 4 1. This is a right triangle problem. Angle A is 90 degrees. Draw the triangle and label it as we did in lecture. If angle B is 64 degrees 15 minutes and side c is 332.98 feet, what is the distance in feet of side b? Give your answer to two decimal places. Do not provide units. Those are in feet – right? 1 points QUESTION 5 1. This is a right triangle problem with angle A being the 90 degree angle. It should look like the one from lecture. If angle B is 31 degrees 10 minutes and side c is 312.86 feet, what is the distance to two decimal places of side a? Give your answer to two decimal places. Do not provide units – those are in feet. 1 points QUESTION 6 Ad by Browse Safe | Close 1. It is desired to determine the height of a flagpole. Assuming that the ground is level, an instrument is set up 216.46 feet from the flagpole with its telescope centered 4.92 feet above the ground. The telescope is sighted horizontally to a point 4.92 feet from the bottom of the flagpole and then the angle at the instrument looking to the top of the pole is measured. That angle is 25 degrees 34 minutes. How tall is the flagpole from its base? Give your answer to two decimal places with NO units. 1 points QUESTION 7 1. You are hiking in the mountains. For every 100.00 feet you would be walking horizontally, you have increased your elevation by 4.6 feet. At what grade are you climbing? Give your answer to three decimal places. Hint: Your units will be in ft/ft. 1 points QUESTION 8 1. A grade of 0.4 percent is being considered for a mountain roadway. The elevation at the initial point is 2,054.23 feet and a horizontal distance of 5,758.79 needs to be covered. What is the elevation at the end of the grade? 1 points QUESTION 9 1. A slope distance was measured between two points (A and T) and determined to be 3,307.97 feet. At point A the elevation is 872.17 feet and at point T the elevation is 884.21 feet. What is the horizontal distance between A and T? 1 points

Question no Assignmnet 3 1 206.154167 degrees 2 0.927183855 3 … Read More...
WEEKLY ASSIGNMENT #5 (WOW THAT TOOK A WHILE) NAME: 1. Find the linear approximation of the function f(x; y; z) = p x2 + y2 + z2 at some point to approximate a value of the number p (3:02)2 + (1:97)2 + (5:99)2. 1 2. Consider your favorite function, the Cobb-Douglas production function. P(L;K) = 1:5L:65K:35 modeling the production of the state of Idaho. Over time we discover that capitol is gradually increasing at an approximate rate of 0:02 units per year. If we decide as a group that we are perfectly happy with our production level and would rather have additional vacation time then how much can we decrease labor by each year and keep the same level of production. In how long(rounded up to the nearest year) will we have an additional week of vacation? 2 3. Use the chain rule to find dz dt or dw=dt (a) z = x?y x+2y x = et; y = e?t. (b) w = sin x cos x x = p t; y = 1=t. 4. Use the chain rule to find @z=@t or @z=@s (a) z = (x ? y)5 x = s2t; y = st2 (b) z = er cos  r = st;  = p x2 + y2. 3 5. The temperature at a point (x; y; z) is given by the function T(x; y; z) = 200e?x2?3y2?9z2 where T is measure in C and x; y; z in meters. (a) Find the rate of change of temperature at the point (2;?1; 2) in the direction toward the point (3;?3; 3). (b) In which direction does the temperature increase fastest, and what is that fastest rate? 4 6. Suppose (1; 1) is a critical point of a function f with continuous second derivatives. In each case, what can you say about f. (a) fxx(1; 1) = 4; fxy(1; 1) = 1; fyy(1; 1) = 2 (b) fxx(1; 1) = 4; fxy(1; 1) = 3; fyy(1; 1) = 2 (c) fxx(1; 1) = ?1; fxy(1; 1) = 6; fyy(1; 1) = 1 (d) fxx(1; 1) = ?1; fxy(1; 1) = 2; fyy(1; 1) = ?8 (e) fxx(1; 1) = 4; fxy(1; 1) = 6; fyy(1; 1) = 9 5 Bonus Show that f(x; y) = x2 + 4y2 ? 4xy + 2 has an infinite number of critical points, and for all of them D = 0 at each one. Then show that f has a local (and absolute) minimum at each critical point. 6

WEEKLY ASSIGNMENT #5 (WOW THAT TOOK A WHILE) NAME: 1. Find the linear approximation of the function f(x; y; z) = p x2 + y2 + z2 at some point to approximate a value of the number p (3:02)2 + (1:97)2 + (5:99)2. 1 2. Consider your favorite function, the Cobb-Douglas production function. P(L;K) = 1:5L:65K:35 modeling the production of the state of Idaho. Over time we discover that capitol is gradually increasing at an approximate rate of 0:02 units per year. If we decide as a group that we are perfectly happy with our production level and would rather have additional vacation time then how much can we decrease labor by each year and keep the same level of production. In how long(rounded up to the nearest year) will we have an additional week of vacation? 2 3. Use the chain rule to find dz dt or dw=dt (a) z = x?y x+2y x = et; y = e?t. (b) w = sin x cos x x = p t; y = 1=t. 4. Use the chain rule to find @z=@t or @z=@s (a) z = (x ? y)5 x = s2t; y = st2 (b) z = er cos  r = st;  = p x2 + y2. 3 5. The temperature at a point (x; y; z) is given by the function T(x; y; z) = 200e?x2?3y2?9z2 where T is measure in C and x; y; z in meters. (a) Find the rate of change of temperature at the point (2;?1; 2) in the direction toward the point (3;?3; 3). (b) In which direction does the temperature increase fastest, and what is that fastest rate? 4 6. Suppose (1; 1) is a critical point of a function f with continuous second derivatives. In each case, what can you say about f. (a) fxx(1; 1) = 4; fxy(1; 1) = 1; fyy(1; 1) = 2 (b) fxx(1; 1) = 4; fxy(1; 1) = 3; fyy(1; 1) = 2 (c) fxx(1; 1) = ?1; fxy(1; 1) = 6; fyy(1; 1) = 1 (d) fxx(1; 1) = ?1; fxy(1; 1) = 2; fyy(1; 1) = ?8 (e) fxx(1; 1) = 4; fxy(1; 1) = 6; fyy(1; 1) = 9 5 Bonus Show that f(x; y) = x2 + 4y2 ? 4xy + 2 has an infinite number of critical points, and for all of them D = 0 at each one. Then show that f has a local (and absolute) minimum at each critical point. 6