How to: Regression & Correlation
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The document provides instructions for using Excel to analyze the relationship between two variables, the odometer reading (X) and selling price (Y) of used cars. It describes how to produce a scatter plot and regression line to model the relationship, and how to interpret the results including the slope, intercept, standard error, coefficient of determination (R2), and testing whether there is a significant linear relationship between the variables.
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The document provides information on correlation and linear regression. It defines correlation as the association between two variables and discusses how the correlation coefficient r measures the strength of this linear association. It then discusses:
- Computing r from sample data
- Testing the hypothesis that r = 0 using a t-test
- Computing the linear regression equation and coefficient of determination
- Using the regression equation to make predictions when there is a significant linear correlation
Two examples are then provided to demonstrate computing r from data, testing for a significant correlation, finding the regression equation, and making a prediction.
Lec20
This document discusses arithmetic functions in digital circuits. It begins by introducing iterative combinational circuits and binary adders such as half adders, full adders, and ripple carry adders. It then covers binary subtraction using two's complement representation and signed binary numbers. The document explains various signed number representations including signed magnitude and signed complement. It provides examples of addition and subtraction for signed integers in these representations.
Chapter 01 Basic Principles of Digital Systems
This document provides an overview of digital systems fundamentals, including:
- Analog signals have continuous values while digital signals can only have discrete values (0 or 1).
- Digital electronics uses binary logic levels to represent information, with a high voltage representing 1 and a low voltage representing 0.
- The binary number system uses positional notation to represent numbers using only the digits 0 and 1.
- Digital circuits operate on binary inputs and outputs, with truth tables listing all possible input-output combinations for a logic gate or circuit.
ANSYS Project
This document discusses modeling heat transfer in a tube heat exchanger using analytical and numerical methods. It will use MATLAB and FLUENT software to model the heat transfer process. Governing equations for the heat transfer and fluid flow are presented. Initial and boundary conditions are defined to solve the equations numerically using a 4th order Runge-Kutta method in MATLAB. The goal is to investigate the results from analytical, numerical and simulation approaches to optimize the heat exchanger efficiency.
ENGR 102B Microsoft Excel Proficiency LevelsPlease have your in.docx
ENGR 102B: Microsoft Excel Proficiency Levels
Please have your instructor or TA initial each level as you complete it. If you need additional help, ask the TAs or use the help guide within Excel.
Once you master Excel Levels I through IV, you can note Excel as a skill on your resume!
Please see D2L Content for this week for your Excel Homework assignment (individual), which is due via D2L Dropbox by the due date specified in the D2L News for your section.
If you use a Mac, please be sure to submit your homework in a format that the grader and instructor can open on a PC.
Level I: Basic Functions Initials _______
1. Calculating an Average: Calculate the arithmetic average of the 5 values listed below. Enter the values in cells A2 through A6. Place a descriptive label in cell A1.
3.6, 3.8, 3.5, 3.7, 3.6
First, calculate the average the long way, by summing the values and dividing by 5:
You will enter the following formula into a blank cell to accomplish this:
=(A2+A3+A4+A5+A6)/5
Second, calculate the average using Excel’s AVERAGE( ) function by entering the following formula in a cell:
=AVERAGE(cellrange)
Replace the “cellrange” with the actual addresses in your spreadsheet of the range of cells holding the five values (i.e., for this problem, the cell range is A2:A6).
2. Determining Velocities (in kph): Some friends at the University of Calgary are coming south for spring break. Help them avoid a speeding ticket by completing a velocity conversion worksheet that calculates the conversion from mph to kph in increments of 10 from 10 to 100. A conversion factor you will need is 0.62 miles/km; you will need this factor to convert from miles/hour to km/hour. Place the conversion factor in its own cell and then reference it in your conversion calculations using absolute cell referencing (e.g., $C$2). Refer to the CBT video on Absolute and Relative Cell Referencing from the “Preparation for the Excel Workshop” assignment if you don’t remember how to do this.
Level II: Advanced Functions Initials _______
1. Projectile Motion I: (See following page for Fig. 1 Excel chart) A projectile is launched at the angle 35o from the horizontal with a velocity equal to 30 m/s. Neglecting air resistance and assuming a horizontal surface, determine how far away from the launch site the projectile will land.
To answer this problem, you will need:
1. Excel’s trigonometry functions to handle the 35o angle, and
2. Equations relating distance to velocity and acceleration
When velocity is constant, as in the horizontal motion of our particle (since we’re neglecting air resistance), the distance traveled is simply the initial horizontal velocity times the time of flight:
(Equation 1)
What keeps the projectile from flying forever is gravity. Since the gravitational acceleration is constant, the vertical distance traveled becomes
(Equation 2)
Because the projectile ends up back on the ground, the final value of y is zero (a hor ...
Variable charts
This document summarizes a lecture on control charts for variables. It introduces the x-R and x-s control charts, which are used to monitor processes and detect changes in the mean (x) and variance (R or s) of a variable. It discusses how to establish control limits for these charts, calculate process capability, and interpret various patterns that may indicate non-random variation, including shifts, cycles and stratification. Rational subgrouping is important to ensure independent samples. The document also covers operating characteristics and how control charts can be used to estimate a process's ability to meet specifications under statistical control.
Matrices and row operations 12123 and applications of matrices
This document discusses using loop analysis to solve electrical circuits. It describes the 5 step process: 1) counting loops, 2) choosing independent loop currents, 3) writing Kirchhoff's voltage law for each loop, 4) solving the system of equations to find loop currents, and 5) reconstructing branch currents from loop currents. It then provides an example solution of a circuit with 3 loops and loop currents of 0.245 A, 0.111 A, and 0.117 A.
Simple regression model
1. The document discusses the simple linear regression model and how to derive the regression coefficients using the least squares method.
2. It uses a numerical example to show how to calculate the regression coefficients b1 and b2 by minimizing the sum of squared residuals.
3. The general method is then described for a model with n observations, where the regression coefficients b1 and b2 are the values that minimize the total sum of squared residuals.
100_2_digitalSystem_Chap1 (2).ppt
This document provides an overview of digital systems and information representation. It discusses key topics such as number systems, arithmetic operations, and coding. Digital systems take discrete inputs and outputs and can have state. There are different types including combinational and sequential systems. Information is represented using physical quantities mapped to discrete values, often binary digits. Common number systems like binary, octal, hexadecimal and their arithmetic are covered. Conversion between number systems and different number bases is also explained.
Lecture - 8 MLR.pptx
Multiple linear regression allows modeling of relationships between a dependent variable and multiple independent variables. It estimates the coefficients (betas) that best fit the data to a linear equation. The ordinary least squares method is commonly used to estimate the betas by minimizing the sum of squared residuals. Diagnostics include checking overall model significance with F-tests, individual variable significance with t-tests, and detecting multicollinearity. Qualitative variables require preprocessing with dummy variables before inclusion in a regression model.
Chapter 03
This chapter discusses graphically solving optimization problems with two design variables. Key steps include:
1. Plotting constraint boundaries and identifying the feasible region.
2. Drawing objective function contours through the feasible region.
3. Locating the optimum solution by observing objective function values at different points.
The chapter provides an example problem of maximizing profit from manufacturing two products. The problem is formulated and solved graphically in detail using the key steps. Mathematica is also introduced as a tool for implementing the graphical solution procedure on a computer.
C2 st lecture 12 the chi squared-test handout
Pearson's chi-squared test is used to determine if there is a relationship between two categorical variables. It has the following structure:
1) State the null and alternative hypotheses
2) Calculate the test statistic by finding residuals between observed and expected counts and summing their squares divided by expected values
3) Find the critical value based on degrees of freedom and significance level
4) Reject the null hypothesis if the test statistic exceeds the critical value, concluding the variables are dependent. Otherwise fail to reject, concluding independence.
Three examples are provided to demonstrate applying the chi-squared test to determine dependence between grades and attendance, height and nose size, and weather and season.
Simple linear regression
1) The document discusses simple linear regression using a scatter diagram and data from a study of employees' years of working experience and income.
2) It presents the scatter diagram and shows how to draw a trend line to roughly estimate dependent variable (income) values from the independent variable (years experience).
3) Equations for the least squares linear regression line are provided, including how to calculate the standard error of estimate, which is interpreted as the standard deviation around the regression line.
silo.tips_the-pressure-derivative.pdf
The document summarizes key aspects of pressure derivatives in well testing:
1) The pressure derivative has important diagnostic properties and allows matching field curves to type curves to better analyze flow regimes like radial flow.
2) During radial flow, the logarithmic pressure derivative will be constant, appearing as a horizontal line on a diagnostic plot. During wellbore storage, it will be a straight line with a slope of one.
3) For buildup testing, the pressure derivative can also identify flow regimes like afterflow or radial flow, appearing as different characteristic lines on a diagnostic plot.
4) Derivatives are approximated numerically, requiring filtering of noise, with algorithms using weighted averages of nearby points to smooth
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How to: Regression & Correlation
- 2. Instructions Ex. 16.1 In Excel: • File/ Open/ Folder:DataSets/ Folder:excel files/ Folder:Ch16/ Xm16-01.xls – To open data file Note: Variable X in the 1st column & variable Y in the 2nd column • Insert/ Chart/ Standard Types: (XY) Scatter/ Next: Specify the Input Y Range & the Input X Range/ Next/ Titles Tab – Title:____; Value (X) axis:____; Value (Y) axis:____; Finish/ - To produce Scatter Diagram • Tools/ Data Analysis / Regression/ OK/ Highlight the Input Y Range & the Input X Range/ Output Options: New Worksheet Ply/ OK - To compute the least squares regression line
- 3. Ex. 16.2
- 4. Ex. 16.2
- 5. Ex. 16.2 Interpretation • The regression line is: ŷ = 17.25 – 0.0669x • The slope coefficient, b1= -0.0669, means that for each additional 1,000 miles on the odometer, the price decreases by an average of $0.0669 thousand, i.e. each additional mile, price decreases by 6.69 cents. • The intercept, b0 = 17.25, means that when the car was not driven at all, the selling price is $17.25 thousand @$17,250 – most probably meaningless!
- 6. Ex. 16.2 Assessing the model 1. Standard Error of Estimate: SSE = 0, when all the points fall on the regression line – thus, smaller SSE excellent fit! SSE =0.3265, compared with y-bar = 14.841, considered small!
- 7. Ex. 16.2 Assessing the model 2. Testing the Slope: Step 1: H0: β1 = 0; No linear relationship (slope =0) H1: β1 =/ 0 Linear relationship exist Step 2: Student t distribution with Degrees of freedom, ν= n -2; Step 3: Test Statistic for β1 (formula) @ b1 ± tα/2sb1 b1 = -13.44 with p-value≈0 (very small). Step 4: There is significance evidence to infer that a linear relationship exist. Step 5: The odometer reading may affect the selling price of cars.
- 8. Ex. 16.2 Assessing the model • Define: Coefficient of Determination - a measure of the strength of the linear relationship: R2 = 0.6483 • It means, 64.83% of the variation in the selling prices is explained by the variation in the odometer readings. The remaining 37.17% is unexplained. • In general, the higher the value of R2, the better the model fits the data.
- 9. Cause & Effect: Coefficient of Correlation • • • • Population coefficient of correlation, ρ (rho) Sample, r ( -1< r <1) Formula: Tools/ Data Analysis Plus/ Correlation (Pearson)/ Variable 1 Range/ Variable 2 Range/ OK
- 10. CORRELATION • r = -0.8052 • H0: ρ = 0; No linear relationship • H1: ρ =/ 0;
- 12. Data source: Managerial Statistics, 9th Ed. (Keller) CENGAGE