GEOG306 Spring 2023 Homework_7

GEOG 306 Spring 2023 Homework 7: Regression Analysis Instructor: Dr. Naijun Zhou Important Instructions  Due: 9AM, Wed., May 3, 2023 on ELMS .  You can write your answers in this Word document, and submit this Word document with your answers on ELMS.  For questions that ask you to do a calculation by hand , you need to show the steps of the calculation . You may use a calculator but you cannot use the calculator’s statistical functions and you cannot use R . If you do not show your calculations, you will not receive any points.  For questions that ask you to do a calculation in R, you need to show your final answer AND the R code that you used to answer the question . Submit ‘clean’ code that is ready to run right away (no results or error messages). If you do not show the R code that you used, you will not receive any points .  Although you won’t get bonus points, it’s highly recommended to type all the formula with a tool (e.g., the Equation in Word). See the lecture slides for examples of typed formula, such as X = ∑ i = 1 n x i n . One of the key tasks in metropolitan area transportation planning is to estimate the total number of trips made by households during a typical day. It is hypothesized that one of the most important determinants of household trips is household income. 12 households are randomly selected to collect their trips per day and household income, and the data are provided in the following table. Trips per day Household Income ($1,000) 3 25 5 36 5 33 4 27 6 42 7 48 9 63 7 54 8 66 8 72 6 51 5 45 Question 1. (1 point). Use software R to plot a scatter gram of trips and household income. Attach the scatter gram to this question. Do you think a linear regression will explain the relationship between the two variables? -1-

GEOG 306 Spring 2023 Homework 7: Regression Analysis Instructor: Dr. Naijun Zhou Yes, I think it will be able to explain the relationship since the data appears to follow a linear pattern between the independent and dependent variables and will be able to determine a line of best fit for the data to show the general trend that the data conveys. Question 2. (1.5 points). Do a regression analysis “by hand” to determine the cause-effect relation between trips and household income. You only need to determine the dependent and independent variables, and calculate the least square regression line (slope and intercept) and r 2 . Independent variable(x)=trips per day and dependent variable(y)=household income b= n ( xiyi ) −( xiyi ) n ( xi 2 ) −( xi ) 2 = b= 12 ( 3696 ) −( 73 ∗ 562 ) 12 ∗ 479 −( 73 ) 2 = 3326 419 = 7.94 a = yi − b ∗ xi n = 562 − 7.94 ∗ 73 12 =− 1.47 y = a + bx = ¿ y=-1.47+7.94x r^2= ∑ ( yihat − ybar ) 2 ∑ ( yi − ybar ) 2 = 2201.29/2537.68 = 0.87 A linear relationship exists with r^2=0.87 Question 3. (1.5 points). Use software R to plot a scatter gram of the independent variable and corresponding residual. Attach the scatter gram to this question. Based on the scatter gram and other information, discuss the residual pattern in terms of the four aspects illustrated in the lecture. -2-

GEOG 306 Spring 2023 Homework 7: Regression Analysis Instructor: Dr. Naijun Zhou -4-