homework 10 key1 - (Q=ID MST.MR.MRE.01.0010l‘ Suppose you...

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Unformatted text preview: (Q=ID: MST.MR.MRE.01.0010l‘) Suppose you work for a firm that is measuring the efi’ect of prioe and advertizing expense on sales and developing a multiple regression model. You would say that: price and advertizing are dependent variables while sales is an independent variable sales, price and advertizing are all independent variables sales, price and advertizing are all dependent variables J . price and advertizing are independent variables while sales is a dependent variable {Q-ID: MST.MR.MRE.01.0020‘U Consider the following excerpt from a multiple regression model for sales (v) with two independent variables: advertizing expense (x3) and prioe (x9). df Regression 2 9589082. 032 4794541 1. 149512 Residual 69 28?}‘94698 4170938 Total 1 297383780 _ Coefficients Standard Error 1: Stat probability-value 4328.74 1485.51 Advertlzing 0.02 0.03 0.54 0.59 Price 33.52 58.96 1.42 0.16 The regression equation is: $ = 1435.51 + 0.03::a + 58.96xp A y = 2.91 + 0an + 1.42xp 4' . $ = 4328.74 + 0.02xa + 33.5219, §= 0.00 + 0.59xa + 0.16xp (Q=ID: M5T.MR.MRE.DI.0030f) Consider the multiple regresslon equatlon: §= 45.3 + 3.4A - 2.25 You can say that: holding B constant, v will decrease by 3.4 for a unlt Increase In A 4 - holding A constant, § will decrease by 2.2 for a unit increase in B holding A constant, v will increase by 2.2 for a unit Increase In 3 § will only increase if the value of A is larger than the value of B (Q=ID: msrnemeomoaor) Multiple regression is used to predict the value of: 4 - a dependent variable based on two or more Independent variables two or more dependent variables based on two or more independent variables an independent variable based on two or more dependent variables two or more independent variables based on a dependent variable (Q-ID: MST.MR.MRE.01.0050F) Consider the multiple regression equation: §= 4523.73 4- 72.42A - 223.3” You can say that: y, A and N are all dependent variables y, A and N are all independent variables y is an independent variable and A and N are dependent variables «I - y is a dependent variable and A and N are independent variables (Q=ID: MST.MR.MRE.01.0060‘|') Consider the following excerpt from a multiple regression model that analyzes the number of visitors to a mountain resort relative to the height ofa mountain and the snow fall on the mountain. — Coefficients Standard Error tStat probability-value Intercept 690.1756757 163.5085998 4.221036 0.0000728 Mountain Height (ft) I 0.029392301 0.099642804 0.294977 0.768897 Snow Fall (in) —3.15833333 5.129175263 -0.61576 0.54008 Select the statement that does not followr from this data: as mountain height increases so does the predicted number of visitors as snow fall increases the predicted number of visitors tends to decrease J - the average number of visitors to a mountain resort is approximately 690 mountain height and snow fall are considered independent variables (Q-ID: M5T.MR.MRE.01.m70f) Consider the following exoerpt from a multiple regression model that analyzes the number of visitors to a theme park based on the number of rides and the price of admission. —m Select the entry fee and number of rides that would produoe the largest predicted number of visitors: $35 entry fee for 50 rides 4' - $80 entry fee for 80 rldes no entry fee for 30 rides $20 entry fee for 40 rldes (Q- ID: MST.MR.1'PII.01.WIOB The coefiicient of multiple determination (R2) is equal to: the total sum of squares multiplied by the regression sum of squares the regression sum of squares multiplied by the total sum of squares J - the regression sum of squares divided by the total sum of squares the total sum of squares divided by the regression sum of squares (Q=ID: MST.MR.TM.01.0020‘F} In multiple regression the F statistic of an overall F test is equal to: the regression mean square divided by the error mean square the error mean square divided by the total mean square X — the regression mean square divided by the total mean square the error mean square divided by the regression mean square (Q-ID: M5T.MR.TM.01.0030F} In multiple regression, the ooefl'lclent of multiple determination (R2) measures the: degree to which each independent variable affects y «I . proportion of the variation in y that is predicted by the set of independent variables proportion of the variation in y that is predicted by one of independent variables degree to which the independent variables are correlated (Q-ID: M5T.MR.TM.01.0040f} Consider the following excerpt from a multiple regression analysis. Significance F _ df 55 MS F Regression 1471103883 3353.519 0.233005 0.392707 Residual 2173338339 31570.13 3'1 2193055.??5 The value of the ooefi'lclent of multiple deterrnlnation (R2) Is: 0.2331 0.006? X - 0.7927 0.9933 (Q-ID: M5T.MR.TM.01.0060f} You are conducting residual analysis upon a multiple regression model that has two independent variables. Select the residual plot that would not be used: residuals versus 9 residuals versus x2 residuals versus x1 J . residuals versus 1/ (Q=ID: M5T.MR..TM.01.0CI?DI') Select the residual plot that is most likeliir to be a plot of residuals versus the predicted value of v: ‘IIJ 0| Hal ID Residuals Residuals t 2 2 ' (U in t :‘l :I , E :9 g 8 5% ° D: I: e 3 PUD!“ j 1 of3 ID: MST.MR.MRE.01.0010 A multiple linear regression model is to be constructed to identify a relationship between a dependent variable (v) and two hdependent variables (x; and xfl. A random sample of an n has been collected and the values 0f Jul. X2. and vi for i = 1. 2. ..., n have been reoorded. 'Ihe residuals (ed in this analysis are defined as the difference between the observed values ot v and the values of v predicted by the regression equation. Select all the assumptions required 1.0 construct a void multiple linear regression model: normalityr of residuals the relationship between the dependent and independent variables is quadratic if Independence of residuals w/ the relationship between the dependent and Independent variables is linear constant variance of residuals J the Independent variables are independent of the dependent variable [1 out at 2] You are partly correct. - Ihe relation-Mp helmeen Ihe dependent and Independent "fl-bl- h Ilneer: you are correct. - Impendenoe dtuldlalll: you are correct. - normality of Mills-ll: this option should have been selected. - eonlllnt vlrlanoe e'l reeldnlil: this option should have been selected. - Ike Independent VI‘I'IIbIBI are lndepuldent efflle dependustvarh‘ble: this is not correct. the relationship between the dependent and independent variables is linear: you are correct. Independence of reeldoale: you are correct. normality or residuals: this option should have been selected. constant varllnce of realthale: this option should have been selected. the Independent variables are Independent ofuie dependent variable: this is not correct. The following assumptions are required to carry out a valid multiple linear regression analysis. Most of the assumptions are related to the raidual tem'is (all. It is Important to notice that the values of the residual terms are unknown until the regression analysis Is conducted. Therefore, it is only possible to check the assumptions of a regression analysis liter the model has been constructed. Linearity between Independent and dependent variables For a multiple linear regression analysis, it is assumed that the relationship between the independent and dependent variables is linear. The regression equation that is oonstmoted to represent the relationship between x1, x2 and y is based on the equation for a plane (a two dimensional flat surface). If the relationship between thee independent and dependent var-labia is not linear- then a plane equation will not provide aocuA-ate predictions of y. A way to tat the assumption or linearity between the Independent and dependent variables Is to plot the values of el against the values of y] and observe whether the residuals are randomly scattered. You should also plot e4 against 3:“ and e; against x1. and observe whether the ruiduais In thee plots are also randomly scattered. If the residuals are not randomly scattered then the assumption of linearity may be violated. Independence of residuals The assumption of hdependence cf rdduais requires that the value of ej and ek should be Independent forj e k. A wayr to tat the assumption of Independence of residuals is to plot a; against i and observe whether there is an identifiable pattern. if there is a pattern then the assumption of independence of residuals may be violated. Normality of residuals The assumption of nonnalty of residuals requires that the values ore; for i = 1, 2, m, n folow a normal distribution. This assumption is made so that the distribution of the predicted values of y can be known (which will be useful for making lnferenoes with the model]. A wayr to test the assumption of normality of residuals Is to construct a normal probability plot forthe valua of e. and observe whether this plot follows an upward sloping straight line. Constant VEHIIIO! d reelduele The assumption of constant variance of residuals requlrfi that the variability of the valua of y will be the same for all values of X; and 22.111} assumption may also be referred to as the assumption of homosoedastidty. A way to tat for homosoedastidty is to plot the values of e. against the values of y} (for several samples) and observe whether the residuals are evenly scattered around zero (which Is the mean of the residuals). if the residuals are not evenly scattered around zero then the assumption of homoscedasticty may be violated. TIZ Df3 ID: MST.MR.MRE.03.0010 Geoff is about to open a mun-ant. He has found two possible locations for the restaurant; one closer to the Central Bus‘nas District (CBD) and one in the suburbs. Geofi is concerned that If he mooses the CED location, he will have to charge hlgher prices for his food to cover the higher rent. Geofi Is uncertain of the afiect that this will have on his annual revenue. To help with his decision, Geolf will carry out a multiple regression analyse to predict the annual revenue earned by restaurants based on their distance from the can in miles and the average price in dollars charged for a dish. Geoff has collected a sample of 50 restaurants that are randomly spread around the city. The pmimlty to the CBD (“10. average price of a dish (x29 and annual revenues (yr) were recorded for eadi ofthese restaurants (I = 1. 2. .... 50). Geoff beleves that a multiple linear regression is app'opriate for this data. After carrying out some analysis, geofi has noticed that moving one mile away from the CED (assuming that the average price of a dish is held constant) will resut In a decrease of $60,000 in annual revenues. Geofi has also noticed that increasing the average price of a dish by $1 (assuming that the distance from the CED is held mnatent) will result in an inn-ease oi $40,000 in annual revenues. The chta shows that a rmtaurant at the center of the can that gave away its food for free would have an annual loss of $20,000. Select the multiple linear regression equation that corresponds to Geoi‘i's analysis: .1. - 20,000 + 50,000“ - 40,000le g. - 20,000 - 40,0001“. + emporium - \‘ri = 40,000 - 60,000x1l + 40.000x2. fl Vi I ‘20.000 4- 4U.WOX1| - 60.000xzi -| Feedback [1 out of 1] You are m. .PPFF!!!9'.!..__.__.__.__. _. _. _. .. .. .. ._ ._ ._ _. _. _. .. .. .. .. ._ ._ ._ _. _. _. .. .. .. ._ ._ ._ _. _. _. .. .. .. ._ ._ ._ _. _. _. .. .. .. .. ._ ._ ._ _. _. _. .. .. .. ._ ._ ._ :| Feedback [1 out of 1] Multiple lineu- regruslon is used Do predict a dependent vu'lable based on the values of two or more Independent variables. It Is assumed that the relations-lip between the dependent and Independent variable ls linear. The general form of the multiple linear regression equation oorrupondlng to Geoff‘s analysis ls: W A VI - L"ti + blxll + b2K2i The data collected by Geoff alowed hlm to determine the values ofthe regression coeiicienis in the mutipie Inear regression equation (that is, bu. b1 and b2). The value on represents the predicted value ol'vr when in. = o and X2: = 0. Since a restaurant at the center of the cab (in; = 0) that gave awav Its food For Free (Km = (I) would have an annual loss of $20.000. the vane of be Is -20.00l]. The value b; represents the slope of y when 3:; Is held constant. Since moving one mlle away From the CBD (assuming that the average price of a dish is held constant) will result in a decrease of $60,000 in annual revenues, the value of b; is "50,000. The value b; represents the slope of R when x; is held constant. Shoe Increasing the average price of a dish by $1 (mmhg that the dlstance from the CEO is held constant) will resul: in an increase of $40,000 in annual revenues, the value of I); Is 40,000. §ubatltuting the value ofthae parameters into the general form ofa multiple regression equation Iwith two independent variables gives: u a 40,000 - sopoox" + 40,0003“. : 3 of 3 ID: MST.MR.MRE.DL001CII3 A multiple linear regrasion model is to be constructed to determine if there Is a relationship between a dependent variable (v) and two Independent variables (:1 and X213 random sample ol'size n has been collected and the values cl'x“, X2; and vi tori = l. 2. .... n have been recorded. The residuals (e.) in tnls analysis are defined as the difference between the observed values of y and the values of y predicted by the regression equation. Select the oondition that is one of the assumptions of a valid multlple linear regression model: the residuals are independent - the Independent variables are Independent of the dependent variable the residuals are constant the relationship between the dependent and independent variables is quadratic Feedback [0 out of 1] This Is not correct. A condition that is one afthe assumptions of a valid multlple linear reg-aslon model Is: the rellrluall are Independent The assumption of Independence of residuals requires that the values ore; and as. should be Independent forj a: k. a way no test the assumption of independence of residuals is to plot e. against i and observe whether there Is an identifiable pattern. Ifthere Is a pattern then the assumption of Independence of residuals may be violated. :i 1 MB ID: MST.MR.MRE.04.0030 Eastpac Bank has constructed a multiple regression equation to represent the relationship between the probability:I that a person will default on theIr mortgage (y) and the age (xi) and annual income (32;) ol‘ that person. The ages In the sample used to oonstmct the equation ranged from 25 to 55 and the annual Incomes ranged from $25,000 per annum to $55,000 per annum. The multiple eASTPAC regression equation corresponding to Eastsz analysis is: 9. - acorn-rs + o.ooax1. - o.oooootoax2. The predicted probability ofdefault on a mortgage for a 36 year old person who earns $34,000 per annum is: 0.093905 0.090405 . 0.092305 0.037505 3 Feedback [0 out or 1] 111ls Is not correct. The predicted probability of def-out on a mtgage for a 36 year old person who earns $34,000 per annum is 0.090405. calculation .. .. .. .. .. .. .. .. A multiple Ihear regression equation (an be used to prediot values are dependent variable for partioilar values oflndependent variables. The predioted probability ofdefauiton a mortgage fora 36 year old person who earns $34,000 per annum can beoaiouiated using the following Formula: W A Yi - b0 + bixii + b2K2i = '0.DS41?5 ‘i- (0.005 X 36) - (0.00000163 X 34,000) - 0.090405 : 3 oi 3 ID: M5T.MR.MRE.04.0010b Macrohard have conducted a multiple linear regression analysis to predict the loading time (y) in milliseconds (thousandths of seconds) for Macrohard workstation files based on the size or the file (X!) In kilobytes and the speed of the processor used to View the file [X2] in megahertz. The analysis was based on a random sample of 400 Macrohard Workstation users.111e file sizes In the sample ranged from 110 to 5,000 kilobytes and the speed of the processors In the sample ranged from 500 megahertz to 4,000 megahertz. 111e mutIple linear regression equation corresponding to Macrohard's analysis is: A la = 233.60 + 0.40!“ - 0.040!” notion-ding to Mao-ohard’s multiple regression equation, it is moot reasonable to oondude that: a 50 kiloby'oe file is predicted to take 233.9 milliseconds to load on a 410 megahertz processor a 5,600 kilobyte file is predicted to take 2,185.6 milliseconds to load on a 6,000 megaheriz processor - holding X2 constant, a one kllobybe increase In in will result In an increase of 0.4 in the predicted value of y holding x2 constant, a one kilobybe increase in x1 will result in a decrease of 0.048 in the predicted value of y - Feedback [1 out of 1] You are correct. Ill-swirled. .. .. . A muitble linear regression equation can be used to predict values of a dependent variable for particular values of independent m (as long as those patioular values of the independent m are within the range oftne sample collected]. The values oftne regression oodiiolanis also provide information about how a ohange in one independent variable will affect the dependent variable (assuming all other variables remain constant). The general form ofthe linear multiple regresson equation corresponding to Mamnard's analysis ls: W ;i = lit) + btxti + bzxzi Holdhg x; oohstant, a one kllobyte Increase In x; will raut In an Increase of0.4 In the predicted value of y slnoe bl = 0.4. I 3 of 3 ID: MST.MR.MRE.01.0010h A muitlole linear regresdon model is to be constructed to deterrnlne If there is a relationship between a dependent variable iv) and two independent verleblee [x1 and :2). A random sample of size n has been collected and the values at x“, xzi and vi fori = 1, 2, ..., n have been recorded. The residuals [en in this analysis are defined as the dll'l'erence between the observed values of v and the values 01' v predlooed by the regresslon equation. select the oondluon that I! one N the assumptions of a valid multlple liner remission model: the relationship between the dependent and independent variables is quadratic the Independent variables are Independent of the dependent variable - the residuals have a constant variance die residuals are constant — Feedback [1 out of 1] Youerewrrect. Piss-wisp. ._ . ._ . ._ . ._ . ._ . ._ . ._ . ._ . ._ . ._ . ._ . ._ . ._ . ._ . ._ . ._ . ._ . ._ . .__ 111e assumption «constant variance of residuals requires that the variablitiir of the values of y will be the same for all values of x1 and x2. This assumption may also be referred to as the assumptlon ofhornoscedasbicltv. a way on to: for hol'rioscedasioltt;r Is to plot the values of q agaln: the values of v. (for several samples) and observe whether the residuals are evenly scattered around zero (url'IIoh B the mean of the reslmele).1fthe residuals are not evenly scattered around we then the mmptlon ofhorrnecedes‘ttdtv may be vlolated. A 3 of3 ID: MST.MR.HRE.O3.0DZD Todd Boaden is a professional skateboarder. He is interested in determining the relationship between the height of his jumps (v) in indies, the number of slices of toast that he eats each morning for breakfast (x1) and the number of hours that he spends skateboarding each delir (xfl. Todd would llloe to construct a multiple Ilnear regresdon equation that he can use to predict at based on x; and x;. Select all the quantities that Todd will reoul'e to predict a value for v using a multiple linear regression equation: a value of x2 within the range of those sampled v’ the 'r intercept a value of x1 wlchln the range of those sampled the mean value of x; J the slope of y with respect to 3:1 when x215 held constant the mean value of v J the slope of y with respect to x2 when x1 is held constant the mean value of x1 3 Feedback [1 out of 2] You are partly correct. the slope ofy with rupectbo :1 when x: is held constant: we are correct. the slope ofv with moo 3:; when x; to hold content: you are correct. a value of :1 within tile range of Blue sampled: thi...
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