PlotCycloidArc(8.5) Math98 HW4 The cylcoid is the plane curve traced out by a point on a circle as the circle rolls without slipping on a straight line.1 In this exercise we will use MATLAB to create an animation of a circle rolling on a straight line, while a point on the circle traces the cycloid. a. Implement a MATLAB function of the form function PlotCycloidArc(ArcLength). This function takes in a positive number ArcLength and displays a snapshot of the cirle rolling (without slipping) on the x-axis while a point on the cirlce traces the cycloid. The circle is initially centered at (0,1) and has radius 1, and the initial tracing point is taken to be (0, 0). An example output is shown in the above ?gure. As in the ?gure, plot the cycloid arc black, the circle blue, and use a red dot for the tracing point. Hint: If the circle has rolled for a length of arc t = 0, the coordinates of the tracing point are (t-sin t, 1-cos t). b. Implement a MATLAB function of the form function CycloidMovie(NumHumps,NumIntervals). This function will output an animation of the circle rolling along a line while a point on the circle traces the cycloid. This function inputs two natural numbers NumHumps and NumIntervals representing the number of periods (or humps) of the cycloid and the number or frames per hump that will be used to make the animation, respectively. Use the getframe command to save frames outputted from PlotCycloidArc and the movie command to play them back together as a movie. Use the axis command to view the frames on the rectan- gle [0, 2pNumHumps] × [0, 5/2]. Also label the ticks 0, 2p, . . . , 2pNumHumps on the x axis with the strings 1See Wikipedia for more on the cycloid. 0, 2p, . . . , 2pNumHumps and do the same for 1, 2 on the y axis (see the ?gure above). Label the movie ’Cycloid Animation’. Submit MATLAB code for both parts a and b and a printout the ?gures obtained by the commands PlotCycloidArc(8.5), PlotCycloidArc(12), and CycloidMovie(3,10)

PlotCycloidArc(8.5) Math98 HW4 The cylcoid is the plane curve traced out by a point on a circle as the circle rolls without slipping on a straight line.1 In this exercise we will use MATLAB to create an animation of a circle rolling on a straight line, while a point on the circle traces the cycloid. a. Implement a MATLAB function of the form function PlotCycloidArc(ArcLength). This function takes in a positive number ArcLength and displays a snapshot of the cirle rolling (without slipping) on the x-axis while a point on the cirlce traces the cycloid. The circle is initially centered at (0,1) and has radius 1, and the initial tracing point is taken to be (0, 0). An example output is shown in the above ?gure. As in the ?gure, plot the cycloid arc black, the circle blue, and use a red dot for the tracing point. Hint: If the circle has rolled for a length of arc t = 0, the coordinates of the tracing point are (t-sin t, 1-cos t). b. Implement a MATLAB function of the form function CycloidMovie(NumHumps,NumIntervals). This function will output an animation of the circle rolling along a line while a point on the circle traces the cycloid. This function inputs two natural numbers NumHumps and NumIntervals representing the number of periods (or humps) of the cycloid and the number or frames per hump that will be used to make the animation, respectively. Use the getframe command to save frames outputted from PlotCycloidArc and the movie command to play them back together as a movie. Use the axis command to view the frames on the rectan- gle [0, 2pNumHumps] × [0, 5/2]. Also label the ticks 0, 2p, . . . , 2pNumHumps on the x axis with the strings 1See Wikipedia for more on the cycloid. 0, 2p, . . . , 2pNumHumps and do the same for 1, 2 on the y axis (see the ?gure above). Label the movie ’Cycloid Animation’. Submit MATLAB code for both parts a and b and a printout the ?gures obtained by the commands PlotCycloidArc(8.5), PlotCycloidArc(12), and CycloidMovie(3,10)

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Matlab project Note: Your final project must be uniquely different from anyone else’s including different from the example I posted!! You are NOT allowed to work together because this is your individual final project!! Anyone caught working together or with similar data/answers will get an automatic zero and will be reported to the Dean’s Office!! REQUIREMENTS %%0. Make a main m-file that you use to run and call your function file. Give it a unique name. Make sure and include your name, your section, and date at the top of the m-file. Suppress any extraneous info; only output what is useful and what follows the intent of your program. (8 points) %%1. Create and use at least one anonymous function somewhere in your program. (5 points) %%2. Make a useful function m-file. That is, create and use at least one user-defined function Use comments immediately below the function definition line that describe what the function does and its inputs and outputs. (10 points) %%3. Utilize proper coding and documentation practices. Comment throughout both the main m-file and the function m-file. Create at least one section (cells). (12 points) %%4. Create and use either one subfunction or one nested function within your function mfile, (10 points) %%5. Use some type of numerical approximation technique like Runge Kutta, Euler’s method, Midpoint Rule, some type of numeric series, etc., 10 pts %%6. Create and use at least one loop (for/while/midpoint), 10 pts %%7. Create and use at least one conditional statement, 10 pts %%8. Create at least one plot, including a title and axes labels at a minimum, 10pts %%9. Output an organized display of your values to a text file that can be opened outside of MATLAB. Include headings so that the display makes sense. 10pts Note: Project need not be fancy or overcomplicated. You want to make sure it runs, meets all the listed requirements, is well-commented and is YOUR OWN WORK !! DELIVERABLES:!! %%10. Submit the following files onto blackboard (ZIP them!): 1. A flowchart or pseudocode of your program plan, 5pts 2. Your main project m-file, 0 credit if not included! 3. Your function m-file, -50% if not included!

Matlab project Note: Your final project must be uniquely different from anyone else’s including different from the example I posted!! You are NOT allowed to work together because this is your individual final project!! Anyone caught working together or with similar data/answers will get an automatic zero and will be reported to the Dean’s Office!! REQUIREMENTS %%0. Make a main m-file that you use to run and call your function file. Give it a unique name. Make sure and include your name, your section, and date at the top of the m-file. Suppress any extraneous info; only output what is useful and what follows the intent of your program. (8 points) %%1. Create and use at least one anonymous function somewhere in your program. (5 points) %%2. Make a useful function m-file. That is, create and use at least one user-defined function Use comments immediately below the function definition line that describe what the function does and its inputs and outputs. (10 points) %%3. Utilize proper coding and documentation practices. Comment throughout both the main m-file and the function m-file. Create at least one section (cells). (12 points) %%4. Create and use either one subfunction or one nested function within your function mfile, (10 points) %%5. Use some type of numerical approximation technique like Runge Kutta, Euler’s method, Midpoint Rule, some type of numeric series, etc., 10 pts %%6. Create and use at least one loop (for/while/midpoint), 10 pts %%7. Create and use at least one conditional statement, 10 pts %%8. Create at least one plot, including a title and axes labels at a minimum, 10pts %%9. Output an organized display of your values to a text file that can be opened outside of MATLAB. Include headings so that the display makes sense. 10pts Note: Project need not be fancy or overcomplicated. You want to make sure it runs, meets all the listed requirements, is well-commented and is YOUR OWN WORK !! DELIVERABLES:!! %%10. Submit the following files onto blackboard (ZIP them!): 1. A flowchart or pseudocode of your program plan, 5pts 2. Your main project m-file, 0 credit if not included! 3. Your function m-file, -50% if not included!

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Paper 1. Narrative Essay Overview: This first paper will be a narrative; in other words, it will be a story. As such it will have these essential items: characters, dialogue, plot, tension, and setting. You will write a story that can be fictional or autobiographical. Format: The first draft will be three typed pages, and you will bring to class 2 copies. It should have your name on every page. Ideally, it will be in MLA (Modern Language Association) format, though this is not important at that stage. See the format of the paper in the example below. Details: For your narrative, you must present a scenario wherein your character, or characters, must deal with and overcome adversity. Consider the essay “Have a Caltastic Day” as an example. In that essay, Streeter tells a success story—of a young man who springs from humble beginnings, overcomes difficult obstacles, and advances his place in the world. In your story, you too will write a story of a person who has faced difficulty. It can be any number of issues that your protagonist faces: a coming of age story involving school, friends, sports, family hardship, etc. You must have a well-developed character. You must have a plot, a clear setting, and use at least some dialogue. Again, it can be based on true events or entirely a work of the imagination. Assessment: In this story, I am looking for well-formed, clear sentences, unified and coherent paragraphs, as well as use of standard grammar, diction, and mechanics of American English. Superior essays will have a clear plot, descriptive language and have material arranged with good supporting details. Sample Paper Format Last Name 1 Your Full Name Dr. Riley-Brown ENG 110: Composition Narrative #1–Draft #1 Date Title of Paper Centered This is where the first line of your paper will go. Double space beneath your title and indent the first line of each paragraph five (5) spaces. The essay should have margins that are one each on each side. You should use Times New Roman font in 12 point font size.

Paper 1. Narrative Essay Overview: This first paper will be a narrative; in other words, it will be a story. As such it will have these essential items: characters, dialogue, plot, tension, and setting. You will write a story that can be fictional or autobiographical. Format: The first draft will be three typed pages, and you will bring to class 2 copies. It should have your name on every page. Ideally, it will be in MLA (Modern Language Association) format, though this is not important at that stage. See the format of the paper in the example below. Details: For your narrative, you must present a scenario wherein your character, or characters, must deal with and overcome adversity. Consider the essay “Have a Caltastic Day” as an example. In that essay, Streeter tells a success story—of a young man who springs from humble beginnings, overcomes difficult obstacles, and advances his place in the world. In your story, you too will write a story of a person who has faced difficulty. It can be any number of issues that your protagonist faces: a coming of age story involving school, friends, sports, family hardship, etc. You must have a well-developed character. You must have a plot, a clear setting, and use at least some dialogue. Again, it can be based on true events or entirely a work of the imagination. Assessment: In this story, I am looking for well-formed, clear sentences, unified and coherent paragraphs, as well as use of standard grammar, diction, and mechanics of American English. Superior essays will have a clear plot, descriptive language and have material arranged with good supporting details. Sample Paper Format Last Name 1 Your Full Name Dr. Riley-Brown ENG 110: Composition Narrative #1–Draft #1 Date Title of Paper Centered This is where the first line of your paper will go. Double space beneath your title and indent the first line of each paragraph five (5) spaces. The essay should have margins that are one each on each side. You should use Times New Roman font in 12 point font size.

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For the second problem of this assignment we’ll try out a fun and useful statistics software package. It’s called “R,” is professional quality, and is available for free on multiple platforms at http://www.r-project.org. Install it on your computer. Note, however, that I didn’t get everything to work right on Linux, so you might want to try Windows or Mac OS/X. We’ll now do linear regression and plotting in 3D using R and R-Commander, as follows: Here’s the really fun part with a rotating 3D graph! Install R-Commander using the menu system in R. Also install the Scatterplot3D package. Make a new dataset using the first 10 rows of Table of chip wirebond pull strength on page 13 in the book. Then ask for a 3D scatter plot of the new data, and include a multiple linear regression with the pull strength as the dependent variable. Submit a screen shot of the graph. Comment on the goodness of fit.

For the second problem of this assignment we’ll try out a fun and useful statistics software package. It’s called “R,” is professional quality, and is available for free on multiple platforms at http://www.r-project.org. Install it on your computer. Note, however, that I didn’t get everything to work right on Linux, so you might want to try Windows or Mac OS/X. We’ll now do linear regression and plotting in 3D using R and R-Commander, as follows: Here’s the really fun part with a rotating 3D graph! Install R-Commander using the menu system in R. Also install the Scatterplot3D package. Make a new dataset using the first 10 rows of Table of chip wirebond pull strength on page 13 in the book. Then ask for a 3D scatter plot of the new data, and include a multiple linear regression with the pull strength as the dependent variable. Submit a screen shot of the graph. Comment on the goodness of fit.

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ENGR 3300: Fluid Mechanics, Fall 2015 Assignment 3 Due: Friday, Oct. 2, 2015 Topics: Chapter 3 & 4 Solutions must be neatly written and must include the following steps (if applicable) to receive full credit. 1. Given: List all known parameters in the problem. 2. Find: List what parameters the problem is asking you to find. 3. Solution: List all equations needed to solve the problem, and show all your work. Draw any necessary sketches or free body diagrams. Circle or box your final answer, and make sure to include appropriate units in your final answer. Grading: 15 total points (10 points for completeness + 5 points for one randomly chosen problem graded for correctness) 1. Water flows at a steady rate up a vertical pipe and out a nozzle into open air. The pipe diameter is 1 inch and the nozzle diameter is 0.5 inches. (a) Determine the minimum pressure that would be required at section 1 (shown in the figure below) to produce a fluid velocity of 30 ft/s at the nozzle (section 2). (b) If the pipe was inverted, determine the minimum pressure that would be required at section 1 to maintain the 30 ft/s velocity at the nozzle. 2. Water flows from a large tank through a small pipe with a diameter of 5 cm. A mercury manometer is placed along the pipe. Assuming the flow is frictionless, (a) estimate the velocity of the water in the pipe and (b) determine the rate of discharge (i.e. volumetric flow rate) from the tank. 3. An engineer is designing a suit for a race car driver and wants to supply cooling air to the suit from an air inlet on the body of the race car. The air speed at the inlet location must be 65 mph when the race car is traveling at 230 mph. Under these conditions, what would be the static pressure at the proposed inlet location? 4. Air flows downward toward a horizontal flat plate. The velocity field is given by ? = (??! − ??!)(2 + cos ??) where a = 5 s-1, ω = 2π s-1, and x and y (measured in meters) are horizontal and vertically upward, respectively, and t is in seconds. (a) Obtain an algebraic equation for a streamline at t = 0. (b) Plot the streamline that passes through point (x,y) = (3,3) at this instant.

ENGR 3300: Fluid Mechanics, Fall 2015 Assignment 3 Due: Friday, Oct. 2, 2015 Topics: Chapter 3 & 4 Solutions must be neatly written and must include the following steps (if applicable) to receive full credit. 1. Given: List all known parameters in the problem. 2. Find: List what parameters the problem is asking you to find. 3. Solution: List all equations needed to solve the problem, and show all your work. Draw any necessary sketches or free body diagrams. Circle or box your final answer, and make sure to include appropriate units in your final answer. Grading: 15 total points (10 points for completeness + 5 points for one randomly chosen problem graded for correctness) 1. Water flows at a steady rate up a vertical pipe and out a nozzle into open air. The pipe diameter is 1 inch and the nozzle diameter is 0.5 inches. (a) Determine the minimum pressure that would be required at section 1 (shown in the figure below) to produce a fluid velocity of 30 ft/s at the nozzle (section 2). (b) If the pipe was inverted, determine the minimum pressure that would be required at section 1 to maintain the 30 ft/s velocity at the nozzle. 2. Water flows from a large tank through a small pipe with a diameter of 5 cm. A mercury manometer is placed along the pipe. Assuming the flow is frictionless, (a) estimate the velocity of the water in the pipe and (b) determine the rate of discharge (i.e. volumetric flow rate) from the tank. 3. An engineer is designing a suit for a race car driver and wants to supply cooling air to the suit from an air inlet on the body of the race car. The air speed at the inlet location must be 65 mph when the race car is traveling at 230 mph. Under these conditions, what would be the static pressure at the proposed inlet location? 4. Air flows downward toward a horizontal flat plate. The velocity field is given by ? = (??! − ??!)(2 + cos ??) where a = 5 s-1, ω = 2π s-1, and x and y (measured in meters) are horizontal and vertically upward, respectively, and t is in seconds. (a) Obtain an algebraic equation for a streamline at t = 0. (b) Plot the streamline that passes through point (x,y) = (3,3) at this instant.

ELEC153 Circuit Theory II M2A4 Lab: AC Parallel Circuits Introduction In this experiment we work with AC parallel circuits. As we did in the AC series circuits lab, the results obtained through Transient Analysis in MultiSim will be verified by manual calculations. Procedure 1. Figure 1 is the circuit we want to analyze.The voltage source is 24 volts peak at 1000 Hz. Figure 1: AC parallel circuit used for analysis using MultiSim Unlike the series circuit, there is no resistor in series with the voltage source that allows us to plot the current by taking advantage of its in-phase relationship. So, in order to measure the current produced by the source (total current) add a 1 Ohm resistor in series with the source. This small resistor will not affect the calculations. Figure 2: Arrangement for analyzing the current waveforms 2. Run the simulations and with the oscilloscope measure both the source voltage and the voltage across the resistor. You should get a plot similar to the following graph: Figure 3: Source voltage (red) and source current (blue) waveforms 3. From the resulting analysis plot, determine the peak current. Record it here. Measured Peak Current 4. Determine the peak current by calculation. Record it here. Does it match the measured peak current? Explain. Calculated Peak Current 5. Calculate the phase-shift. Using the method presented in the last lab, measure the time difference at the zero-crossing of the two signals. Record it here. Time difference 6. From the resulting calculation, determine the phase shift by using the following formula Record it here. Measured Phase Shift 7. Determine the phase shift by calculation. Record it here. Does it match the measured phase shift? Explain. Calculated Phase Shift 8. Change the frequency of the voltage source to 5000 Hz. Re-simulate and perform a Transient Analysis to find the new circuit current and phase angle. Measure them and record them here: Measured Current Measured Phase Shift 9. Perform the manual calculations needed to find the circuit current and phase shift. Record the calculated values here. Do they match the measured values within reason? What has happened to the circuit with an increase in frequency? Calculated Current Calculated Phase Shift 10. Replace the capacitor with a 0.8 H inductor. Set the source frequency back to 1000 Hz. Perform Transient Analysis and measure the current amplitude and phase shift. Record them here: Measured Current Measured Phase Shift 11. Perform the manual calculations needed to find the circuit current and phase shift. Record the calculated values here. Do they match the measured values within reason? Calculated Current Calculated Phase Shift 12. Change the frequency of the voltage source to 5000 Hz. Re-simulate and perform a Transient Analysis to find the new circuit current and phase angle. Measure them and record them here: Measured Current Measured Phase Shift 13. Perform the manual calculations needed to find the circuit current and phase shift. Record the calculated values here. Do they match the measured values within reason? What has happened to the circuit with an increase in frequency? Calculated Current Calculated Phase Shift Write-up and Submission In general, for each lab you do, you will be asked to setup certain circuits, simulate them, record the results, verify the results are correct by hand, and then discuss the solution. Your lab write-up should contain a one page, single spaced discussion of the lab experiment, what went right for you, what you had difficulty with, what you learned from the experiment, how it applies to our coursework, and any other comment you can think of. In addition, you should include screen shots from the MultiSim software and any other figure, table, or diagram as necessary.

ELEC153 Circuit Theory II M2A4 Lab: AC Parallel Circuits Introduction In this experiment we work with AC parallel circuits. As we did in the AC series circuits lab, the results obtained through Transient Analysis in MultiSim will be verified by manual calculations. Procedure 1. Figure 1 is the circuit we want to analyze.The voltage source is 24 volts peak at 1000 Hz. Figure 1: AC parallel circuit used for analysis using MultiSim Unlike the series circuit, there is no resistor in series with the voltage source that allows us to plot the current by taking advantage of its in-phase relationship. So, in order to measure the current produced by the source (total current) add a 1 Ohm resistor in series with the source. This small resistor will not affect the calculations. Figure 2: Arrangement for analyzing the current waveforms 2. Run the simulations and with the oscilloscope measure both the source voltage and the voltage across the resistor. You should get a plot similar to the following graph: Figure 3: Source voltage (red) and source current (blue) waveforms 3. From the resulting analysis plot, determine the peak current. Record it here. Measured Peak Current 4. Determine the peak current by calculation. Record it here. Does it match the measured peak current? Explain. Calculated Peak Current 5. Calculate the phase-shift. Using the method presented in the last lab, measure the time difference at the zero-crossing of the two signals. Record it here. Time difference 6. From the resulting calculation, determine the phase shift by using the following formula Record it here. Measured Phase Shift 7. Determine the phase shift by calculation. Record it here. Does it match the measured phase shift? Explain. Calculated Phase Shift 8. Change the frequency of the voltage source to 5000 Hz. Re-simulate and perform a Transient Analysis to find the new circuit current and phase angle. Measure them and record them here: Measured Current Measured Phase Shift 9. Perform the manual calculations needed to find the circuit current and phase shift. Record the calculated values here. Do they match the measured values within reason? What has happened to the circuit with an increase in frequency? Calculated Current Calculated Phase Shift 10. Replace the capacitor with a 0.8 H inductor. Set the source frequency back to 1000 Hz. Perform Transient Analysis and measure the current amplitude and phase shift. Record them here: Measured Current Measured Phase Shift 11. Perform the manual calculations needed to find the circuit current and phase shift. Record the calculated values here. Do they match the measured values within reason? Calculated Current Calculated Phase Shift 12. Change the frequency of the voltage source to 5000 Hz. Re-simulate and perform a Transient Analysis to find the new circuit current and phase angle. Measure them and record them here: Measured Current Measured Phase Shift 13. Perform the manual calculations needed to find the circuit current and phase shift. Record the calculated values here. Do they match the measured values within reason? What has happened to the circuit with an increase in frequency? Calculated Current Calculated Phase Shift Write-up and Submission In general, for each lab you do, you will be asked to setup certain circuits, simulate them, record the results, verify the results are correct by hand, and then discuss the solution. Your lab write-up should contain a one page, single spaced discussion of the lab experiment, what went right for you, what you had difficulty with, what you learned from the experiment, how it applies to our coursework, and any other comment you can think of. In addition, you should include screen shots from the MultiSim software and any other figure, table, or diagram as necessary.

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MAE 384. Advanced Mathematical Methods for Engineers. The army is interested in characterizing the acoustic signature of a helicopter. The following data show measurements of acoustic pressure (made dimensionless) for a two-bladed helicopter rotor through 1 2 of a rotor revolution. The data points are equally spaced in time, and the period of the data collection is 1 6 of a second. p0 = [ 0 0.0004 0.0015 0.0028 0.0040 0.0048 0.0057 0.0071 0.0095 0.0134 . . . 0.0185 0.0242 0.0302 0.0364 0.0447 0.0577 0.0776 0.0955 0.0907 -0.0477 . . . -0.0812 -0.0563 -0.0329 -0.0127 0.0032 0.0147 0.0221 0.0256 0.0255 0.0222 . . . 0.0170 0.0112 0.0064 0.0035 0.0023 0.0020 0.0019 0.0016 0.0009 0.0002 ] (a) Find the real Discrete Fourier Transform for this data set. That is, …nd the Fourier coe¢ cients (the Ak’s and Bk’s). (b) Any term in the Fourier series can be written: Ak cos(k!t) + Bk sin(k!t) = Ck cos(k!t + k) where Ck = q A2 k + B2 k and k = tan?1 ?Bk Ak Find the Ck’s and plot their amplitude vs. k to illustrate the relative size of each term in the series. (The amplitude should drop of with increasing k.) (c) Plot the function (Fourier series) and the original data on the same plot. (d) The actual loudness of the helicopter depends on the maximum peak-to-peak amplitude of the signal. Find the peak-to-peak amplitude by …nding the maximum and minimum values of p0 as predicted by the Fourier series solution. Recall that a function has a maximum or a minimum when its derivative equals zero. (e) Extra Credit. Try …nding Ak’s and Bk’s for k > N 2 (where N = the number of data points). Show that the resulting series does not represent the data.

MAE 384. Advanced Mathematical Methods for Engineers. The army is interested in characterizing the acoustic signature of a helicopter. The following data show measurements of acoustic pressure (made dimensionless) for a two-bladed helicopter rotor through 1 2 of a rotor revolution. The data points are equally spaced in time, and the period of the data collection is 1 6 of a second. p0 = [ 0 0.0004 0.0015 0.0028 0.0040 0.0048 0.0057 0.0071 0.0095 0.0134 . . . 0.0185 0.0242 0.0302 0.0364 0.0447 0.0577 0.0776 0.0955 0.0907 -0.0477 . . . -0.0812 -0.0563 -0.0329 -0.0127 0.0032 0.0147 0.0221 0.0256 0.0255 0.0222 . . . 0.0170 0.0112 0.0064 0.0035 0.0023 0.0020 0.0019 0.0016 0.0009 0.0002 ] (a) Find the real Discrete Fourier Transform for this data set. That is, …nd the Fourier coe¢ cients (the Ak’s and Bk’s). (b) Any term in the Fourier series can be written: Ak cos(k!t) + Bk sin(k!t) = Ck cos(k!t + k) where Ck = q A2 k + B2 k and k = tan?1 ?Bk Ak Find the Ck’s and plot their amplitude vs. k to illustrate the relative size of each term in the series. (The amplitude should drop of with increasing k.) (c) Plot the function (Fourier series) and the original data on the same plot. (d) The actual loudness of the helicopter depends on the maximum peak-to-peak amplitude of the signal. Find the peak-to-peak amplitude by …nding the maximum and minimum values of p0 as predicted by the Fourier series solution. Recall that a function has a maximum or a minimum when its derivative equals zero. (e) Extra Credit. Try …nding Ak’s and Bk’s for k > N 2 (where N = the number of data points). Show that the resulting series does not represent the data.

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Statistical Methods (STAT 4303) Review for Final Comprehensive Exam Measures of Central Tendency, Dispersion Q.1. The data below represents the test scores obtained by students in college algebra class. 10,12,15,20,13,16,14 Calculate (a) Mean (b) Median (c) Mode (d) Variance, s2 (e) Coefficient of variation (CV) Q.2. The data below represents the test scores obtained by students in English class. 12,15,16,18,13,10,17,20 Calculate (a) Mean (b) Median (c) Mode (d) Variance, s2 (e) Coefficient of variation (CV) (f) Compare the results of Q.1 and Q.2, Which scores College Algebra or English do you think is more precise (less spread)? Q.3 Following data represents the score obtained by students in one of the exams 9, 13, 14, 15, 16, 16, 17, 19, 20, 21, 21, 22, 25, 25, 26 Create a frequency table to calculate the following descriptive statistics (a) mean (b) median (c) mode (d) first and third quartiles (e) Construct Box and Whisker plot. (f) Comment on the shape of the distribution. (g) Find inter quartile range (IQR). (h) Are there any outliers (based on IQR technique)? In the above problem, if the score 26 is replaced by 37 (i) What will happen to the mean? Will it increase, decrease or remains the same? (j) What will be the new median? (k) What can you say about the effect of outliers on mean and median? Q.4 Following data represents the score obtained by students in one of the exams 19, 14, 14, 15, 17, 16, 17, 20, 20, 21, 21, 22, 25, 25, 26, 27, 28 Create a frequency table to calculate the following descriptive statistics a) mean b) median c) mode d) first and third quartiles e) Construct Box and Whisker plot. f) Comment on the shape of the distribution. g) Find inter quartile range (IQR). h) Are there any outliers (based on IQR technique)? In the above problem, if the score 28 is replaced by 48 i) What will happen to the mean? Will it increase, decrease or remains the same? j) What will be the new median? k) What can you say about the effect of outliers on mean and median? Q.5 Consider the following data of height (in inch) and weight(in lbs). Height(x) Frequency 50 2 52 3 55 2 60 4 62 3  Find the mean height.  What is the variance of height? Also, find the standard deviation. (c) Find the coefficient of variation (CV). Q.6. The following table shows the number of miles run during one week for a sample of 20 runners: Miles Mid-value (x) Frequency (f) 5.5-10.5 1 10.5-15.5 2 15.5-20.5 3 20.5-25.5 5 25.5-30.5 4 (a) Find the average (mean) miles run. (Hint: Find mid-value of mile range first) (b) What is the variance of miles run? Also, find the standard deviation. (c) Find the coefficient of variation (CV). Q.7. (a) If the mean of 20 observations is 20.5, find the sum of all observations? (b) If the mean of 30 observations is 40, find the sum of all observations? Probability Q.8 Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. a) How many students are in both classes? b) What is the probability that a randomly-chosen student from this group is taking only the Chemistry class? Q.9 A drawer contains 4 red balls, 5 green balls, and 5 blue balls. One ball is taken from the drawer and then replaced. Another ball is taken from the drawer. What is the probability that (Draw tree diagram to facilitate your calculation). (a) both balls are red (b) first ball is red (c) both balls are of same colors (d) both balls are of different colors (e) first ball is red and second ball is blue (f) first ball is red or blue Q.10 A drawer contains 3 red balls, 5 green balls, and 5 blue balls. One ball is taken from the drawer and not replaced. Another ball is then taken from the drawer. Draw tree diagram to facilitate your calculation. What is the probability that (a) both balls are red (b) first ball is red (c) both balls are of same colors (d) both balls are of different colors (e) first ball is red and second ball is blue (f) first ball is red or blue Q. 11 Missile A has 45% chance of hitting target. Missile B has 55% chance of hitting a target. What is the probability that (i) both miss the target. (ii) at least one will hit the target. (iii) exactly one will hit the target. Q. 12 A politician from D party speaks truth 65% of times; another politician from rival party speaks truth 75% of times. Both politicians were asked about their personal love affair with their own office secretary, what is the probability that (i) both lie the actual fact . (ii) at least one will speak truth. (iii) exactly one speaks the truth. (iv) both speak the truth. Q.13 The question, “Do you drink alcohol?” was asked to 220 people. Results are shown in the table. . Yes No Total Male 48 82 Female 24 66 Total (a) What is the probability of a randomly selected individual being a male also drinks? (b) What is the probability of a randomly selected individual being a female? (c) What is the probability that a randomly selected individual drinks? (d) A person is selected at random and if the person is female, what is the probability that she drinks? (e) What is the probability that a randomly selected alcoholic person is a male? Q.14 A professor, Dr. Drakula, taught courses that included statements from across the five colleges abbreviated as AH, AS, BA, ED and EN. He taught at Texas A&M University – Kingsville (TAMUK) during the span of five academic years AY09 to AY13. The following table shows the total number of graduates during AY09 to AY13. One day, he was running late to his class. He was so focused on the class that he did not stop for a red light. As soon as he crossed through the intersection, a police officer Asked him to stop. ( a ) It is turned out that the police officer was TAMUK graduate during the past five years. What is the probability that the Police Officer was from ED College? ( b ) What is the probability that the Police Officer graduated in the academic year of 2011? ( c ) If the traffic officer graduated from TAMUK in the academic year of 2011(AY11). What is the conditional probability that he graduated from the ED college? ( d ) Are the events the academic year “AY 11” and the college of Education “ED” independent? Yes or no , why? Discrete Distribution Q.15 Find k and probability for X=2 and X=4. X 1 2 3 4 5 P(X=x) 0.1 3k 0.2 2k 0.2 (Hint: First find k, and then plug in) Also, calculate the expected value of X, E(X) and variance V(X). A game plan is derived based on above table, a player wins $5 if he can blindly choose 3 and loses $1 if he chooses other numbers.What is his expected win or loss per game? If he plays this game for 20 times, what is total win or lose? Q.16 Find k. X 3 4 5 6 7 P(X=x) k 2k 2k k 2k (Hint: First find k, and then plug in) Also, calculate the expected value of X, E(X) and variance V(X). A game plan is derived based on above table, a player wins $5 if he can blindly choose 3 and loses $1 if he chooses other numbers. What is his expected win or loss per game? If he plays this game for 20 times, what is total win or lose? Binomial Distribution: Q.17 (a) Hospital records show that of patients suffering from a certain disease, 75% die of it. What is the probability that of 6 randomly selected patients, 4 will recover? (b) A (blindfolded) marksman finds that on the average he hits the target 4 times out of 5. If he fires 4 shots, what is the probability of (i) more than 2 hits? (ii) at least 3 misses? (c) which of the following are binomial experiments? Explain the reason. i. Telephone surveying a group of 200 people to ask if they voted for George Bush. ii. Counting the average number of dogs seen at a veterinarian’s office daily. iii. You take a survey of 50 traffic lights in a certain city, at 3 p.m., recording whether the light was red, green, or yellow at that time. iv. You are at a fair, playing “pop the balloon” with 6 darts. There are 20 balloons. 10 of the balloons have a ticket inside that say “win,” and 10 have a ticket that says “lose.” Normal Distribution Q.18 Use standard normal distribution table to find the following probabilities: (a) P(Z<2.5) (b) P(Z< -1.3) (c) P(Z>0.12) (d) P(Z> -2.15) (e) P(0.11<Z<0.22) (f) P(-0.11<Z<0.5) Q.19. Use normal distribution table to find the missing values (?). (a) P(Z< ?)=0.40 (b) P(Z< ?)=0.76 (c) P(Z> ?)=0.87 (d) P(Z> ?)=0.34 Q.20. The length of life of certain type of light bulb is normally distributed with mean=220hrs and standard deviation=20hrs. (a) Define a random variable, X A light bulb is randomly selected, what is the probability that (b) it will last will last more than 207 hrs. ? (c) it will last less than 214 hrs. (d) it will last in between 199 to 207 hrs. Q.21. The length of life of an instrument produced by a machine has a normal distribution with a mean of 22 months and standard deviation of 4 months. Find the probability that an instrument produced by this machine will last (a) less than 10 months. (b) more than 28 months (c) between 10 and 28 months. Distribution of sample mean and Central Limit Theorem (CLT) Q.22 It is assumed that weight of teenage student is normally distributed with mean=140 lbs. and standard deviation =15 lbs. A simple random sample of 40 teenage students is taken and sample mean is calculated. If several such samples of same size are taken (i) what could be the mean of all sample means. (ii) what could be the standard deviation of all sample means. (iii) will the distribution of sample means be normal ? (iv) What is CLT? Write down the distribution of sample mean in the form of ~ ( , ) 2 n X N   . Q.23 The time it takes students in a cooking school to learn to prepare seafood gumbo is a random variable with a normal distribution where the average is 3.2 hours and a standard deviation of 1.8 hours. A sample of 40 students was investigated. What is the distribution of sample mean (express in numbers)? Hypothesis Testing Q.24 The NCHS reported that the mean total cholesterol level in 2002 for all adults was 203 with standard deviation of 37. Total cholesterol levels in participants who attended the seventh examination of the Offspring in the Framingham Heart Study are summarized as follows: n=3,00, =200.3. Is there statistical evidence of a difference in mean cholesterol levels in the Framingham Offspring (means does the result form current examination differs from 2002 report)?? (Follow the steps below to reach the conclusion) (i) Define null and alternate hypothesis (Also write what is  , and x in words at the beginning) (ii) Identify the significance level ,  and check whether it is one sided or two sided test. (iii) Calculate test statistics, Z. (iv) Use standard normal table to find the p-value and state whether you reject or accept (fail to reject) the null hypothesis. (v) what is the critical value, do you reject or accept the H0. (vi) Write down the conclusion based on part (iv). Q.25 A sample of 145 boxes of Kellogg’s Raisin Bran contain in average 1.95 scoops of raisins. It is known from past experiments that the standard deviation for the number of scoops of raisins is 0.25. The manufacturer of Kellogg’s Raisin Bran claimed that in average their product contains more than 2 scoops of raisins, do you reject or accept the manufacturers claim (follow all five steps)? Q.26 It is assumed that the mean systolic blood pressure is μ = 120 mm Hg. In the Honolulu Heart Study, a sample of n = 100 people had an average systolic blood pressure of 130.1 mm Hg. The standard deviation from the population is 21.21 mm Hg. Is the group significantly different (with respect to systolic blood pressure!) from the regular population? Use 10% level of significance. Q.27 A CEO claims that at least 80 percent of the company’s 1,000,000 customers are very satisfied. Again, 100 customers are surveyed using simple random sampling. The result: 73 percent are very satisfied. Based on these results, should we accept or reject the CEO’s hypothesis? Assume a significance level of 0.05. Q.28 True/False questions (These questions are collected from previous HW, review and exam problems, see the previous solutions for answers) (a) Total sum of probability can exceed 1. (b) If you throw a die, getting 2 or any even number are independent events. (c) If you roll a die for 20 times, the probability of getting 5 in 15th roll is 20 15 . (d) A student is taking a 5 question True-False quiz but he has not been doing any work in the course and does not know the material so he randomly guesses at all the answers. Probability that he gets the first question right is 2 1 . (e) Typing in laptop and writing emails using the same laptop are independent events. (f) Normal distribution is right skewed. (g) Mean is more robust to outliers. So mean is used for data with extreme values. (h) It is possible to have no mode in the data. (i) Standard normal variable, Z has some unit. (j) Only two parameters are required to describe the entire normal distribution. (k) Mean of standard normal variable, Z is 1. (l) If p-value of more than level of significance (alpha), we reject the H0. (m) Very small p-value indicates rejection of H0. (n) H0 always contains equality sign. (o) CLT indicates that distribution of sample mean can be anything, not just normal. (p) Sample mean is always equal to population mean. (q) Variance of sample mean is less than population mean. (r) Variance of sample mean does not depend on sample size. (s) Mr. A has cancer but a medical doctor diagnosed him as “no cancer”. It is a type I error. (t) Level of significance is probability of making type II error. (u) Type II error can be controlled. (v) Type I error is more serious than type II error. (w) Type I and Type II errors are based on null hypothesis. Q.29 Type I and Type II Errors : Make statements about Type I (False Positive) and Type II errors (False Negative). (a) The Alpha-Fetoprotein (AFP) Test has both Type I and Type II error possibilities. This test screens the mother’s blood during pregnancy for AFP and determines risk. Abnormally high or low levels may indicate Down syndrome. (Hint: Take actual status as down syndrome or not) Ho: patient is healthy Ha: patient is unhealthy (b) The mechanic inspects the brake pads for the minimum allowable thickness. Ho: Vehicles breaks meet the standard for the minimum allowable thickness. Ha: Vehicles brakes do not meet the standard for the minimum allowable thickness. (c) Celiac disease is one of the diseases which can be misdiagnosed or have less diagnosis. Following table shows the actual celiac patients and their diagnosis status by medical doctors: Actual Status Yes No Diagnosed as celiac Yes 85 5 No 25 105 I. Calculate the probability of making type I and type II error rates. II. Calculate the power of the test. (Power of the test= 1- P(type II error) Answers: USEFUL FORMULAE: Descriptive Statistics Possible Outliers, any value beyond the range of Q 1.5( ) and Q 1.5( ) Range = Maximum value -Minimum value 100 where 1 ( ) (Preferred) 1 and , n fx x For data with repeats, 1 ( ) (Preferred ) OR 1 and n x x For data without repeats, 1 3 1 3 3 1 2 2 2 2 2 2 2 2 2 2 Q Q Q Q x s CV n f n f x x OR s n fx nx s n x x s n x nx s                             Discrete Distribution         ( ) ( ) ( ) ( ) { ( )} ( ) ( ) 2 2 2 2 E X x P X x V X E X E X E X xP X x Binomial Distribution Probability mass function, P(X=x)= x n x n x C p q  for x=0,1,2,…,n. E(X)=np, Var(X)=npq Hypothesis Testing based on Normal Distribution      X std X mean Z Standard Normal Variable, Probability Bayes Rule, ( ) ( and ) ( ) ( ) ( | ) P B P A B P B P A B P A B    Central Limit Theorem For large n (n>30), ~ ( , ) 2 n X N   and ˆ ~ ( , ) n pq p N p For hypothesis testing of μ, σ known           n x Z   For hypothesis testing of p n pq p p Z   ˆ ANSWERS: Q.1 (a) 14.286 (b) 14 (c) none (d) 10.24 (e) 22.40 Q.2 (a) 15.125 (b) 15.5 (c) No (d) 10.98 (e) 21.9 (f) English Q.3 (a) 18.6 (b)19 (c) 16, 21, and 25 (d) 15, 22 (f) slightly left (g) 7 (h) no outliers (i) increase (j) same Q.4 (a) 0.41 (b) 20 (c)14, 17, 20, 21,25 (d) 16.5, 25 (f) slightly right (g) 8.5 (h) no (i) increase (j) same Q.5 (a)56.57 (b) 22.26 (c) 8.34 Q.6 (a) 21 (b) 38.57 (c) 29.57 Q.7 (a) 410 (b) 1200 Q.8 (a)3 (b) 0.65 Q.9 (a) 0.082 (b) 0.29 (c)0.34 (d) 0.66 (e)0.10 (f) 0.64 Q.10 (a) 0.038 (b)0.23 (c) 0.71 (d) 0.29 (e)0.096 (f) 0.62 Q.11 (i)0.248 (ii)0.752 (iii)0.505 Q.12 (i)0.0875 (ii)0.913 (iii)0.425 (iii)0.488 Q.13 (a)0.22 (b)0.41 (c)0.33 (d)0.27 (e) 0.67 Q.14 (a) 0.13 (b) 0.18 (c)0.12 Q.15 E(X)=3.1 , V(X)=1.69, $0.2 per game, $ 4 win. Q.16 E(X)=5.125, V(X)=1.86, $0.25 loss per game, $5 loss. Q.17 (a)0.201 (b) 0.819, 0.027 Q.18 (a)0.9938 (b)0.0968 (c)0.452 (d)0.984 (e) 0.0433 (f)0.2353 Q.19 (a) -0.25 (b)0.71 (c) -1.13 (d)0.41 Q.20 (b) 0.7422 (c) 0.3821 (d) 0.1109 Q.21 (a)0.0014 (b) 0.0668 (c) 0.9318 Q.22 (a) 140 (b)2.37 Q.24 Z=-1.26, Accept null. Q.25 Z=-2.41, accept null Q.26 Z=4.76, reject H0 Q.27 Z=-1.75, reject H0 Q.28 F, F, F, T , F, F, F, T, F, T, F, F, T, T, F, F, T, F, T, F, F, T, T Q.29 (c)0.113 , 0.022 , 0.977 (or 98%)

Statistical Methods (STAT 4303) Review for Final Comprehensive Exam Measures of Central Tendency, Dispersion Q.1. The data below represents the test scores obtained by students in college algebra class. 10,12,15,20,13,16,14 Calculate (a) Mean (b) Median (c) Mode (d) Variance, s2 (e) Coefficient of variation (CV) Q.2. The data below represents the test scores obtained by students in English class. 12,15,16,18,13,10,17,20 Calculate (a) Mean (b) Median (c) Mode (d) Variance, s2 (e) Coefficient of variation (CV) (f) Compare the results of Q.1 and Q.2, Which scores College Algebra or English do you think is more precise (less spread)? Q.3 Following data represents the score obtained by students in one of the exams 9, 13, 14, 15, 16, 16, 17, 19, 20, 21, 21, 22, 25, 25, 26 Create a frequency table to calculate the following descriptive statistics (a) mean (b) median (c) mode (d) first and third quartiles (e) Construct Box and Whisker plot. (f) Comment on the shape of the distribution. (g) Find inter quartile range (IQR). (h) Are there any outliers (based on IQR technique)? In the above problem, if the score 26 is replaced by 37 (i) What will happen to the mean? Will it increase, decrease or remains the same? (j) What will be the new median? (k) What can you say about the effect of outliers on mean and median? Q.4 Following data represents the score obtained by students in one of the exams 19, 14, 14, 15, 17, 16, 17, 20, 20, 21, 21, 22, 25, 25, 26, 27, 28 Create a frequency table to calculate the following descriptive statistics a) mean b) median c) mode d) first and third quartiles e) Construct Box and Whisker plot. f) Comment on the shape of the distribution. g) Find inter quartile range (IQR). h) Are there any outliers (based on IQR technique)? In the above problem, if the score 28 is replaced by 48 i) What will happen to the mean? Will it increase, decrease or remains the same? j) What will be the new median? k) What can you say about the effect of outliers on mean and median? Q.5 Consider the following data of height (in inch) and weight(in lbs). Height(x) Frequency 50 2 52 3 55 2 60 4 62 3  Find the mean height.  What is the variance of height? Also, find the standard deviation. (c) Find the coefficient of variation (CV). Q.6. The following table shows the number of miles run during one week for a sample of 20 runners: Miles Mid-value (x) Frequency (f) 5.5-10.5 1 10.5-15.5 2 15.5-20.5 3 20.5-25.5 5 25.5-30.5 4 (a) Find the average (mean) miles run. (Hint: Find mid-value of mile range first) (b) What is the variance of miles run? Also, find the standard deviation. (c) Find the coefficient of variation (CV). Q.7. (a) If the mean of 20 observations is 20.5, find the sum of all observations? (b) If the mean of 30 observations is 40, find the sum of all observations? Probability Q.8 Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. a) How many students are in both classes? b) What is the probability that a randomly-chosen student from this group is taking only the Chemistry class? Q.9 A drawer contains 4 red balls, 5 green balls, and 5 blue balls. One ball is taken from the drawer and then replaced. Another ball is taken from the drawer. What is the probability that (Draw tree diagram to facilitate your calculation). (a) both balls are red (b) first ball is red (c) both balls are of same colors (d) both balls are of different colors (e) first ball is red and second ball is blue (f) first ball is red or blue Q.10 A drawer contains 3 red balls, 5 green balls, and 5 blue balls. One ball is taken from the drawer and not replaced. Another ball is then taken from the drawer. Draw tree diagram to facilitate your calculation. What is the probability that (a) both balls are red (b) first ball is red (c) both balls are of same colors (d) both balls are of different colors (e) first ball is red and second ball is blue (f) first ball is red or blue Q. 11 Missile A has 45% chance of hitting target. Missile B has 55% chance of hitting a target. What is the probability that (i) both miss the target. (ii) at least one will hit the target. (iii) exactly one will hit the target. Q. 12 A politician from D party speaks truth 65% of times; another politician from rival party speaks truth 75% of times. Both politicians were asked about their personal love affair with their own office secretary, what is the probability that (i) both lie the actual fact . (ii) at least one will speak truth. (iii) exactly one speaks the truth. (iv) both speak the truth. Q.13 The question, “Do you drink alcohol?” was asked to 220 people. Results are shown in the table. . Yes No Total Male 48 82 Female 24 66 Total (a) What is the probability of a randomly selected individual being a male also drinks? (b) What is the probability of a randomly selected individual being a female? (c) What is the probability that a randomly selected individual drinks? (d) A person is selected at random and if the person is female, what is the probability that she drinks? (e) What is the probability that a randomly selected alcoholic person is a male? Q.14 A professor, Dr. Drakula, taught courses that included statements from across the five colleges abbreviated as AH, AS, BA, ED and EN. He taught at Texas A&M University – Kingsville (TAMUK) during the span of five academic years AY09 to AY13. The following table shows the total number of graduates during AY09 to AY13. One day, he was running late to his class. He was so focused on the class that he did not stop for a red light. As soon as he crossed through the intersection, a police officer Asked him to stop. ( a ) It is turned out that the police officer was TAMUK graduate during the past five years. What is the probability that the Police Officer was from ED College? ( b ) What is the probability that the Police Officer graduated in the academic year of 2011? ( c ) If the traffic officer graduated from TAMUK in the academic year of 2011(AY11). What is the conditional probability that he graduated from the ED college? ( d ) Are the events the academic year “AY 11” and the college of Education “ED” independent? Yes or no , why? Discrete Distribution Q.15 Find k and probability for X=2 and X=4. X 1 2 3 4 5 P(X=x) 0.1 3k 0.2 2k 0.2 (Hint: First find k, and then plug in) Also, calculate the expected value of X, E(X) and variance V(X). A game plan is derived based on above table, a player wins $5 if he can blindly choose 3 and loses $1 if he chooses other numbers.What is his expected win or loss per game? If he plays this game for 20 times, what is total win or lose? Q.16 Find k. X 3 4 5 6 7 P(X=x) k 2k 2k k 2k (Hint: First find k, and then plug in) Also, calculate the expected value of X, E(X) and variance V(X). A game plan is derived based on above table, a player wins $5 if he can blindly choose 3 and loses $1 if he chooses other numbers. What is his expected win or loss per game? If he plays this game for 20 times, what is total win or lose? Binomial Distribution: Q.17 (a) Hospital records show that of patients suffering from a certain disease, 75% die of it. What is the probability that of 6 randomly selected patients, 4 will recover? (b) A (blindfolded) marksman finds that on the average he hits the target 4 times out of 5. If he fires 4 shots, what is the probability of (i) more than 2 hits? (ii) at least 3 misses? (c) which of the following are binomial experiments? Explain the reason. i. Telephone surveying a group of 200 people to ask if they voted for George Bush. ii. Counting the average number of dogs seen at a veterinarian’s office daily. iii. You take a survey of 50 traffic lights in a certain city, at 3 p.m., recording whether the light was red, green, or yellow at that time. iv. You are at a fair, playing “pop the balloon” with 6 darts. There are 20 balloons. 10 of the balloons have a ticket inside that say “win,” and 10 have a ticket that says “lose.” Normal Distribution Q.18 Use standard normal distribution table to find the following probabilities: (a) P(Z<2.5) (b) P(Z< -1.3) (c) P(Z>0.12) (d) P(Z> -2.15) (e) P(0.11 ?)=0.87 (d) P(Z> ?)=0.34 Q.20. The length of life of certain type of light bulb is normally distributed with mean=220hrs and standard deviation=20hrs. (a) Define a random variable, X A light bulb is randomly selected, what is the probability that (b) it will last will last more than 207 hrs. ? (c) it will last less than 214 hrs. (d) it will last in between 199 to 207 hrs. Q.21. The length of life of an instrument produced by a machine has a normal distribution with a mean of 22 months and standard deviation of 4 months. Find the probability that an instrument produced by this machine will last (a) less than 10 months. (b) more than 28 months (c) between 10 and 28 months. Distribution of sample mean and Central Limit Theorem (CLT) Q.22 It is assumed that weight of teenage student is normally distributed with mean=140 lbs. and standard deviation =15 lbs. A simple random sample of 40 teenage students is taken and sample mean is calculated. If several such samples of same size are taken (i) what could be the mean of all sample means. (ii) what could be the standard deviation of all sample means. (iii) will the distribution of sample means be normal ? (iv) What is CLT? Write down the distribution of sample mean in the form of ~ ( , ) 2 n X N   . Q.23 The time it takes students in a cooking school to learn to prepare seafood gumbo is a random variable with a normal distribution where the average is 3.2 hours and a standard deviation of 1.8 hours. A sample of 40 students was investigated. What is the distribution of sample mean (express in numbers)? Hypothesis Testing Q.24 The NCHS reported that the mean total cholesterol level in 2002 for all adults was 203 with standard deviation of 37. Total cholesterol levels in participants who attended the seventh examination of the Offspring in the Framingham Heart Study are summarized as follows: n=3,00, =200.3. Is there statistical evidence of a difference in mean cholesterol levels in the Framingham Offspring (means does the result form current examination differs from 2002 report)?? (Follow the steps below to reach the conclusion) (i) Define null and alternate hypothesis (Also write what is  , and x in words at the beginning) (ii) Identify the significance level ,  and check whether it is one sided or two sided test. (iii) Calculate test statistics, Z. (iv) Use standard normal table to find the p-value and state whether you reject or accept (fail to reject) the null hypothesis. (v) what is the critical value, do you reject or accept the H0. (vi) Write down the conclusion based on part (iv). Q.25 A sample of 145 boxes of Kellogg’s Raisin Bran contain in average 1.95 scoops of raisins. It is known from past experiments that the standard deviation for the number of scoops of raisins is 0.25. The manufacturer of Kellogg’s Raisin Bran claimed that in average their product contains more than 2 scoops of raisins, do you reject or accept the manufacturers claim (follow all five steps)? Q.26 It is assumed that the mean systolic blood pressure is μ = 120 mm Hg. In the Honolulu Heart Study, a sample of n = 100 people had an average systolic blood pressure of 130.1 mm Hg. The standard deviation from the population is 21.21 mm Hg. Is the group significantly different (with respect to systolic blood pressure!) from the regular population? Use 10% level of significance. Q.27 A CEO claims that at least 80 percent of the company’s 1,000,000 customers are very satisfied. Again, 100 customers are surveyed using simple random sampling. The result: 73 percent are very satisfied. Based on these results, should we accept or reject the CEO’s hypothesis? Assume a significance level of 0.05. Q.28 True/False questions (These questions are collected from previous HW, review and exam problems, see the previous solutions for answers) (a) Total sum of probability can exceed 1. (b) If you throw a die, getting 2 or any even number are independent events. (c) If you roll a die for 20 times, the probability of getting 5 in 15th roll is 20 15 . (d) A student is taking a 5 question True-False quiz but he has not been doing any work in the course and does not know the material so he randomly guesses at all the answers. Probability that he gets the first question right is 2 1 . (e) Typing in laptop and writing emails using the same laptop are independent events. (f) Normal distribution is right skewed. (g) Mean is more robust to outliers. So mean is used for data with extreme values. (h) It is possible to have no mode in the data. (i) Standard normal variable, Z has some unit. (j) Only two parameters are required to describe the entire normal distribution. (k) Mean of standard normal variable, Z is 1. (l) If p-value of more than level of significance (alpha), we reject the H0. (m) Very small p-value indicates rejection of H0. (n) H0 always contains equality sign. (o) CLT indicates that distribution of sample mean can be anything, not just normal. (p) Sample mean is always equal to population mean. (q) Variance of sample mean is less than population mean. (r) Variance of sample mean does not depend on sample size. (s) Mr. A has cancer but a medical doctor diagnosed him as “no cancer”. It is a type I error. (t) Level of significance is probability of making type II error. (u) Type II error can be controlled. (v) Type I error is more serious than type II error. (w) Type I and Type II errors are based on null hypothesis. Q.29 Type I and Type II Errors : Make statements about Type I (False Positive) and Type II errors (False Negative). (a) The Alpha-Fetoprotein (AFP) Test has both Type I and Type II error possibilities. This test screens the mother’s blood during pregnancy for AFP and determines risk. Abnormally high or low levels may indicate Down syndrome. (Hint: Take actual status as down syndrome or not) Ho: patient is healthy Ha: patient is unhealthy (b) The mechanic inspects the brake pads for the minimum allowable thickness. Ho: Vehicles breaks meet the standard for the minimum allowable thickness. Ha: Vehicles brakes do not meet the standard for the minimum allowable thickness. (c) Celiac disease is one of the diseases which can be misdiagnosed or have less diagnosis. Following table shows the actual celiac patients and their diagnosis status by medical doctors: Actual Status Yes No Diagnosed as celiac Yes 85 5 No 25 105 I. Calculate the probability of making type I and type II error rates. II. Calculate the power of the test. (Power of the test= 1- P(type II error) Answers: USEFUL FORMULAE: Descriptive Statistics Possible Outliers, any value beyond the range of Q 1.5( ) and Q 1.5( ) Range = Maximum value -Minimum value 100 where 1 ( ) (Preferred) 1 and , n fx x For data with repeats, 1 ( ) (Preferred ) OR 1 and n x x For data without repeats, 1 3 1 3 3 1 2 2 2 2 2 2 2 2 2 2 Q Q Q Q x s CV n f n f x x OR s n fx nx s n x x s n x nx s                             Discrete Distribution         ( ) ( ) ( ) ( ) { ( )} ( ) ( ) 2 2 2 2 E X x P X x V X E X E X E X xP X x Binomial Distribution Probability mass function, P(X=x)= x n x n x C p q  for x=0,1,2,…,n. E(X)=np, Var(X)=npq Hypothesis Testing based on Normal Distribution      X std X mean Z Standard Normal Variable, Probability Bayes Rule, ( ) ( and ) ( ) ( ) ( | ) P B P A B P B P A B P A B    Central Limit Theorem For large n (n>30), ~ ( , ) 2 n X N   and ˆ ~ ( , ) n pq p N p For hypothesis testing of μ, σ known           n x Z   For hypothesis testing of p n pq p p Z   ˆ ANSWERS: Q.1 (a) 14.286 (b) 14 (c) none (d) 10.24 (e) 22.40 Q.2 (a) 15.125 (b) 15.5 (c) No (d) 10.98 (e) 21.9 (f) English Q.3 (a) 18.6 (b)19 (c) 16, 21, and 25 (d) 15, 22 (f) slightly left (g) 7 (h) no outliers (i) increase (j) same Q.4 (a) 0.41 (b) 20 (c)14, 17, 20, 21,25 (d) 16.5, 25 (f) slightly right (g) 8.5 (h) no (i) increase (j) same Q.5 (a)56.57 (b) 22.26 (c) 8.34 Q.6 (a) 21 (b) 38.57 (c) 29.57 Q.7 (a) 410 (b) 1200 Q.8 (a)3 (b) 0.65 Q.9 (a) 0.082 (b) 0.29 (c)0.34 (d) 0.66 (e)0.10 (f) 0.64 Q.10 (a) 0.038 (b)0.23 (c) 0.71 (d) 0.29 (e)0.096 (f) 0.62 Q.11 (i)0.248 (ii)0.752 (iii)0.505 Q.12 (i)0.0875 (ii)0.913 (iii)0.425 (iii)0.488 Q.13 (a)0.22 (b)0.41 (c)0.33 (d)0.27 (e) 0.67 Q.14 (a) 0.13 (b) 0.18 (c)0.12 Q.15 E(X)=3.1 , V(X)=1.69, $0.2 per game, $ 4 win. Q.16 E(X)=5.125, V(X)=1.86, $0.25 loss per game, $5 loss. Q.17 (a)0.201 (b) 0.819, 0.027 Q.18 (a)0.9938 (b)0.0968 (c)0.452 (d)0.984 (e) 0.0433 (f)0.2353 Q.19 (a) -0.25 (b)0.71 (c) -1.13 (d)0.41 Q.20 (b) 0.7422 (c) 0.3821 (d) 0.1109 Q.21 (a)0.0014 (b) 0.0668 (c) 0.9318 Q.22 (a) 140 (b)2.37 Q.24 Z=-1.26, Accept null. Q.25 Z=-2.41, accept null Q.26 Z=4.76, reject H0 Q.27 Z=-1.75, reject H0 Q.28 F, F, F, T , F, F, F, T, F, T, F, F, T, T, F, F, T, F, T, F, F, T, T Q.29 (c)0.113 , 0.022 , 0.977 (or 98%)

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Linkage analysis: The linkage shown below is the kinematic sketch for the rear suspension of a motorcycle. The dimensions are given in the drawing; consider BC = 2” along the AB direction for the given configuration. For the analysis, assume that the angular velocity of the input link (link 2) is constant and operating in the CW direction (corresponding to the motorcycle going over a bump). a. Calculate the mobility of the linkage. How many loop equations are needed to solve for the dependent joint variables? b. Formulate the loop equations. c. Solve the loop equations and give explicit expressions for the dependent variables as a function of the input angle ?. d. Compute the limits for the input angle ?. Is the linkage going to work as expected? (is the range of motion of ? enough?) e. Write the position vector of point C. Use Maple, GIM or similar software to plot the trajectory of point C over the range of ? calculated in d). f. Use Maple or similar software to plot ? and s as a function of ?. g. Use GIM to create a simulation for the motion of the linkage. Provide a snapshot. h. Compute the velocity vector for point C. Give the value of the velocity for the configuration shown in the kinematic sketch (? = 200o), for an input angular velocity of 200 rpm. i. Compute the velocity s of the slider. Plot this velocity as a function of ? for a constant input angular velocity of 200rpm. j. Plot the acceleration of the slide, s, as a function of ?, for the same constant input angular velocity

Linkage analysis: The linkage shown below is the kinematic sketch for the rear suspension of a motorcycle. The dimensions are given in the drawing; consider BC = 2” along the AB direction for the given configuration. For the analysis, assume that the angular velocity of the input link (link 2) is constant and operating in the CW direction (corresponding to the motorcycle going over a bump). a. Calculate the mobility of the linkage. How many loop equations are needed to solve for the dependent joint variables? b. Formulate the loop equations. c. Solve the loop equations and give explicit expressions for the dependent variables as a function of the input angle ?. d. Compute the limits for the input angle ?. Is the linkage going to work as expected? (is the range of motion of ? enough?) e. Write the position vector of point C. Use Maple, GIM or similar software to plot the trajectory of point C over the range of ? calculated in d). f. Use Maple or similar software to plot ? and s as a function of ?. g. Use GIM to create a simulation for the motion of the linkage. Provide a snapshot. h. Compute the velocity vector for point C. Give the value of the velocity for the configuration shown in the kinematic sketch (? = 200o), for an input angular velocity of 200 rpm. i. Compute the velocity s of the slider. Plot this velocity as a function of ? for a constant input angular velocity of 200rpm. j. Plot the acceleration of the slide, s, as a function of ?, for the same constant input angular velocity

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