You are trying to assess the demand for your new product. The collected historical dataon similar products from the market (listed below). There are fourvariables:

demand:product demand in $100,000

market:market share of the closest competitor (percentage)price:product price in $100.

brand:brand awareness of your product (defined between 0-1).

When needed, you can usea significance level of 5%in answering the following questions.

P1.Estimate the correlations between all four variables. Do you observe any initial concern for multicollinearity?

a)Yes, since most correlation estimates between demand and the other three variables are too high.

b)Yes, since most correlation estimates between market, price, and brand are too high.

c)No, since most correlation estimates between demand and the other three variables are too low.

d)No, since most correlation estimates between market, price, and brand are low.

Use the least squares method to estimate a multiple linear regression model where demand is the dependent variable and the other three (market, price, and brand) are independent variables.

P2.The estimated regression model is given by

a) demand = 9.94-0.97(market) -0.10(price) + 1.58(brand)+ b) demand = 9.94-0.97(market) -0.10(price) + 1.58(brand)

c) demand = 0-0.97(market) -0.10(price) + 1.58(brand) +

d) demand = 0+0.97(market) +0.10(price) -1.58(brand) +

P3.Interpret the meanings of the partial regression coefficient in front of the variable"market"()in the context of the problem.

a) Expected decrease in the demand when the market share of the competitor goes up by one unit.

b) Expected demand when the market share is zero

c) Expected decrease in the demand when the market share of the competitor goes up by one unit given the same level of price and brand awareness.

d) Expected increase in the demand when the market share of the competitor goes up by one unit given the same level of price and brand awareness.

Can you be reasonably certain that price has an effect on demand given the market share of the competitor and the brand awareness?

P4.What type of test you will be using to test the hypothesis? a) t-test b) independent samples t-test c) F-testd) z-test

P5.What is the value of the test statistic? a) 71.02 b) -4.41 c) -19.69 d) 16.49

P6.What is the critical value? a) 0.05 b) 0.10 c) -1.66 d) -1.98

P7.What is the p-value?a) 0 b) 0.10 c) 0.05 d) 0.025

P8.What is your conclusion? a) Reject H0 b) Fail to Reject H0 c) Reject Ha d) Fail to Reject Ha

P9.What is your best estimate ofdemand when the competitor's market share is 50%,the price of your product is $100 and the brand awareness of your product is 0.9. a) -37 b) -47.64 c) 0.81 d) 10.78

To test multi-collinearity, compute the VIF measures for each variable.

P10.The VIF estimate forthe "market"variable isa) 1 b) 10.2 c) 7.69 d) 0.05

P11.The VIF estimate for the "price" variable isa) 1 b) 10.2 c) 7.69 d) 0.05

P12.The VIF estimate for the "brand" variable isa) 1 b) 10.2 c) 7.69 d) 0.05

P13.Is there a multi-collinearity issue in your multiple linear regression model?

a) Yes, as most correlation estimates between demand and the other three variables were too high.

b) Yes, as the VIF estimates are low.

c) No, as the VIF estimates are low.

d) Yes, the R-squared of the multiple linear regression is too high (0.87).

Test if the residuals from your multiple linear regression model are normally distributed.

P14.The test statistic is a) 0.98 b) 0.10 c) 0.18 d) 0.05

P15.The p-value is a) 0.98 b) 0.10 c) 0.18 d) 0.05

P16.Your conclusion is

a) Fail to reject the null hypothesis, the residuals are normally distributed.

b) Fail to reject the null hypothesis, the residuals are not normally distributed. c) Reject the null hypothesis, the residuals are normally distributed.

d) Reject the null hypothesis, the residuals are not normally distributed.

Test if the residuals from your multiple linear regression model exhibit constant variance

P17.The test statistic is a) 0.05 b) 11.07 c) 8.7 d) 16.21

P18.Critical value is a) 0.05 b) 11.07 c) 87 d) 16.91

P19.Your conclusion is

a) Fail to reject the null hypothesis, the residuals exhibit constant variance.

b) Fail to reject the null hypothesis, the residuals do not exhibit constant variance. c) Reject the null hypothesis, the residuals exhibit constant variance.

d) Reject the null hypothesis, the residuals do not exhibit constant variance.

Visually test if the residuals from your multiple linear regression model are correlated

P20.Your conclusion is

a) The residuals are correlated, there is a clear and strong pattern in the"Residuals-Fitted"plot.

b) The residuals are not correlated, there does not appear to be a strong pattern in the"Residuals-Fitted"plot.

c) The residuals are not correlated, as the q-q plot shows a straight line.

d) The residuals are correlated, as the q-q plot shows a straight line.

Data:

demand" "market" "price" "brand"

"1" 9.38154812740516 0.492435918585124 10.1827701937207 0.54737003492678

"2" 9.21436963486873 0.600591410090087 9.53849356229135 0.583531129502478

"3" 9.24467149369725 0.48740334230077 10.4038724248647 0.550962953977248

"4" 9.10372010611702 0.552317114621226 13.3501914049673 0.586453818250611

"5" 9.24931082137373 0.402049025560564 11.5306907188373 0.499578556338294

"6" 9.62796016435587 0.429219636724706 8.92911331308047 0.701665941623344

"7" 9.03958764207756 0.528755418179056 12.2450694854734 0.515860843835919

"8" 9.00107453211844 0.514582923000275 10.3998832835727 0.442708030387185

"9" 9.41563574104207 0.470207028479618 8.97072815830496 0.583132516239036

"10" 9.07513951737279 0.505154013848956 11.5108001268408 0.481572922809269

"11" 9.12653576295872 0.439223119338067 12.2156531016353 0.438263651526648

"12" 9.32315964976076 0.507278297887817 8.31623163540911 0.536062857429897

"13" 9.23007761874421 0.503286128113087 8.085307446738 0.377150111933529

"14" 9.37725193987364 0.54558321163204 8.72422305344938 0.521363528582097

"15" 9.5992806538382 0.528064181609967 5.51568618827725 0.519957541135788

"16" 8.99304602890807 0.552022703202841 13.3498415827337 0.576580101810876

"17" 9.01082854164034 0.541737700462045 12.3303292226018 0.527950026949376

"18" 9.1951913310064 0.511511083809158 12.1819800732279 0.634297158119624

"19" 9.36556017816337 0.535050390991223 10.1651098113221 0.610937876629004

"20" 9.43877374985382 0.538026100924432 10.0897960621848 0.620127534337832

"21" 9.16828077168679 0.464690393053311 8.99269230998558 0.342051707279741

"22" 8.97561902471687 0.550090897517199 11.8411577505828 0.498150211670718

"23" 9.47917351722605 0.408734142306318 10.3655745792905 0.566766252840283

"24" 9.42646245489912 0.528599886121535 9.481221656249 0.553885036489854

"25" 8.96381880933045 0.433201251911005 13.6024022965779 0.480650980728151

"26" 9.46461142751816 0.493171981227958 9.20820620317573 0.432779978995746

"27" 9.12369591099781 0.579845027122428 9.87153594448052 0.411966802305586

"28" 9.1982018091922 0.462459789552277 10.2786390343713 0.456525583197191

"29" 9.11539667513571 0.492021514771433 11.5795253824655 0.609267284665112

"30" 8.97472521320659 0.42800786177065 11.4897820672752 0.349654028737412

"31" 9.09277904030406 0.558157503845854 8.43387441106469 0.310390192815587

"32" 9.23639835592513 0.438986497834625 8.85227969612725 0.403683159246038

"33" 9.62615102183141 0.503816948164135 6.56343797386412 0.408750589333525

"34" 9.52491018071872 0.476324689130228 9.21311526652053 0.62209871966647

"35" 8.76130197826326 0.473568925825635 11.6288238789797 0.299540049930016

"36" 10.0675592101257 0.52015453612041 4.94469101435406 0.690757655090002

"37" 9.37470041147609 0.512375758577876 9.31451966920695 0.563871300235198

"38" 9.10361117662356 0.435114324903567 13.5090169179424 0.502224105564435

"39" 9.09373275986476 0.497367203520874 9.7215483516034 0.388651299019799

"40" 8.66845219830321 0.559976703015056 12.5463437743041 0.329503284689428

"41" 9.13367878386301 0.511306564454956 7.680442670506 0.334223088043614

"42" 9.72966083154834 0.413888814630104 6.30952189661265 0.542105207420131

"43" 9.43600298593889 0.465856786370354 8.46277881367349 0.478153118718174

"44" 9.05237636517156 0.514359538731514 10.6681541862602 0.509729084340511

"45" 9.33498468502116 0.446215329509241 10.8108770955925 0.607808647125592

"46" 9.00933211383341 0.477715314804205 14.3379445871587 0.566513201666651

"47" 8.8221191820621 0.492824635227595 12.0775944960101 0.421630410473266

"48" 8.66522622335771 0.591353327385235 10.7510241825757 0.307435220217294

"49" 9.37110700435898 0.46637360428923 8.47834915180225 0.525854261683057

"50" 9.13442408584075 0.49301838819051 9.27859646242912 0.435229450650554

"51" 9.06546300842403 0.540070296926486 8.84038374986518 0.317478247914102

"52" 9.42450194651559 0.487047412919705 7.97518589260736 0.448425507967312

"53" 9.22597889405974 0.52629971547603 11.3252399764225 0.587531016669337

"54" 9.45481752617944 0.401939472732208 10.5114830603386 0.623519713323226

"55" 9.84872168537666 0.51845938886435 7.7581320984397 0.683275222962607

"56" 9.70687999186756 0.509048025957192 7.49502065120645 0.61681129888555

"57" 9.5529836875261 0.526742594860407 11.3825364271054 0.657918950242874

"58" 9.71515994487481 0.534771502947451 6.68036754648647 0.478014035592425

"59" 9.34436085061778 0.48449816712394 9.53293647316323 0.522308036323329

"60" 9.26936034694 0.444233797689899 11.8520139678821 0.506183065900372

"61" 8.81395826955138 0.468594040638195 14.3715029908767 0.421352044218113

"62" 8.92321381081282 0.583170203072669 10.3342749333267 0.439138947299513

"63" 9.67814957710414 0.556876691063263 7.06980025364584 0.498033520985517

"64" 8.87813816450508 0.538570732524239 11.8793397562603 0.470959135609712

"65" 9.10657636966013 0.497637489801279 11.0695274859161 0.5162308356341

"66" 9.08399624231618 0.494296160882726 8.88185391767593 0.349094031593903

"67" 9.20847192763108 0.4948497028127 11.5117357103252 0.535693500544526

"68" 9.0667879971077 0.547719884405939 11.6225859972443 0.523225305676731

"69" 9.22909475038358 0.495927593404233 9.73628675690566 0.443416737937838

"70" 9.31295459053844 0.486986016370191 8.08566274401315 0.357726193130785

"71" 8.83566402863898 0.482869660412159 9.60296976299444 0.34510343816951

"72" 9.13838734150643 0.510086504392438 11.4767158604547 0.523888103982166

"73" 8.50593607011967 0.568442365536692 11.3319741067316 0.254309002712924

"74" 9.64389054137735 0.480258402241569 6.86163611947207 0.565390053879998

"75" 8.86687442322721 0.476029907802006 13.5749544430043 0.400630004729722

"76" 9.32040928879079 0.551267631345627 8.98146374146043 0.459144690986665

"77" 8.96927869764145 0.52816844678856 10.14203215251 0.383544734884072

"78" 9.71929870185098 0.52412659339023 8.4114861670606 0.597648202606911

"79" 9.12342328504315 0.53041106779947 10.3123122427424 0.448458597355772

"80" 9.29950469266671 0.566304609221213 11.2436138774826 0.63288036056699

"81" 9.00927614115194 0.471315195504711 11.4798836983554 0.445025048003613

"82" 9.40272289270717 0.431469905069235 9.55432913295603 0.586048832073528

"83" 9.12892843444941 0.508303339524865 8.86005278290964 0.442054710309439

"84" 9.33335814438541 0.49816349007175 8.59493707316142 0.4465879921029

"85" 9.17537032358595 0.485430751367345 10.2013672858775 0.473323858851686

"86" 9.33479256530297 0.532470951247558 11.9008446101995 0.652758836111661

"87" 9.08311143733298 0.445190052179477 12.5330569048721 0.625318891440327

"88" 9.08630504502065 0.539668746400784 11.5614318837854 0.604371559757418

"89" 8.88684641679577 0.444478447865233 12.9521493445614 0.493276079554316

"90" 9.12969077224447 0.553596142839898 11.5811515420123 0.598655533435253

"91" 9.70955302366509 0.453749760137529 8.04279777768081 0.72760706048222

"92" 9.32195602820324 0.478553868751674 10.5138255187179 0.545819360906028

"93" 8.78188438043168 0.524796693549241 10.1835991799958 0.311375445422939

"94" 9.13743369201062 0.513802497226937 11.1140336447978 0.43556706908631

"95" 9.492559811981 0.444457608161075 10.8601081593896 0.560805545999396

"96" 8.99299422837146 0.460307772167295 12.6545688928507 0.50430442170307

"97" 9.26697957642626 0.39863080632608 10.804442559601 0.577231923758801

"98" 9.36082282664089 0.514743590761012 9.42470632455214 0.513464549355413

"99" 9.1509083890555 0.500394444267184 12.1793816640845 0.63565051534507

"100" 9.27309535487249 0.530170322577511 9.82724875805924 0.583705260247683