Quiz 3 - stat(1)

Heating Cost Temperature Insulation Garage Garage (Yes/No) SUMMARY OUTPUT 250 35 3 0 No 360 29 4 1 Yes Regression 165 36 7 0 No Multiple R 43 60 6 0 No R Square 92 65 5 0 No Adjusted R Square 200 30 5 0 No Standard Error 355 10 6 1 Yes Observations 290 7 10 1 Yes 230 21 9 0 No ANOVA 120 55 2 0 No 73 54 12 0 No Regression 205 48 5 1 Yes Residual 400 20 5 1 Yes Total 320 39 4 1 Yes 72 60 8 0 No 272 20 5 1 Yes Intercept 94 58 7 0 No Temperature 190 40 8 1 Yes Insulation 235 27 9 0 No Garage 139 30 7 0 No 1. Predicted equation of multiple regression. The multiple regression equation allows us to use more than one independent variable (in this case X1, X2, and X3) to explain the variation one dependent variable (Y) Our predicted equation regression is Y = B0 + B1X1 + B2X2 + B3X3 so plugging in the values we get Y Given that our dependent variable is heating cost, we get that with every one unit increase in tempe 2. Standard error of equation The standard error of the equation is the average distance the observed values fall from the regressio With the standard error of the regression being 41.6184, it means that, on average, the actual obser 3. coefficient of multiple determination The coefficient of multiple determination (R square) tells us how much variance the dependent varia Given that the coefficient of multiple determination is 0.8698 we can say that 86.98% of the variation 4. Significance of overall regression model, including hypothesis Using an alpha of 0.05 and comparing it to our F-significance value of 2.58643977507557E-07 we get that 0.05 > 2.58643977507557E-07 therefore we reject the null hypothesis that there is no re

Thus, you can indeed use this model to determine the value of Y 5. Evaluate individual regression coefficients, including hypothesis Given our model Y = 393.665680544729 -3.96284720942939X1 -11.3339537291218X2 + 77.432104 We get that: .One unit increase in X1 (temperature) will decrease Y by 3.9628 units everything else .One unit increase in X2 (insulation) will decrease Y by 11.334 units everything else he .On average, the heating cost when there is a garage exceeds the heating cost when t 77.4321 units, everything else held constant We then test the following hypothesis in each variable H0: b1 =0 (No relationship exists between X1 and Y) Given the p-value associated with X1 is lower t H1: b1 ≠ 0 (Relationship exists between X1 and Y) We do the same for X2 H0: b1 =0 (No relationship exists between X2 and Y) Given the p-value associated with X2 is lower t H1: b1 ≠ 0 (Relationship exists between X2 and Y) We do the same for X3 H0: b1 =0 (No relationship exists between X2 and Y) Given the p-value associated with X3 is lower t H1: b1 ≠ 0 (Relationship exists between X2 and Y) 6. Confidence intervals of coefficients Confidence interval for X1 Lower level = -5.3464 Upper level = -2.5793 we are 95% confident that one unit increase in X1 will decrease Y between 5.3464 and 2.5793 units. Because this confidence interval does not include zero, we have evidence to conclude that there is a relationship between X1 and Y Confidence interval for X2 Lower level = -19.8168 Upper level = -2.8511 we are 95% confident that one unit increase in X2 will decrease Y between 19.8168 and 2.8511 units. Because this confidence interval does not include zero, we have evidence to conclude that there is a relationship between X2 and Y

n Statistics 0.932651207862495 0.869838275527371 0.845432952188754 41.6184162866806 20 df SS MS F Significance F 3 185202.3 61734.09 35.64133379667 2.59E-07 16 27713.48 1732.093 19 212915.8 Coefficients andard Erro t Stat P-value Lower 95% Upper 95% Lower 95.0% 393.665680544729 45.00128 8.747876 1.707180037E-07 298.267218195 489.0641 298.267218195 -3.96284720942939 0.652657 -6.071864 1.617544109E-05 -5.34641919005 -2.579275 -5.346419190048 -11.3339537291218 4.001531 -2.832404 0.012010209998 -19.8168213738 -2.851086 -19.81682137383 77.4321045649189 22.78282 3.398706 0.003670176949 29.1346797774 125.7295 29.13467977738 NOTE All values, with the exception of the coefficients, are rounded to 4 decimal spaces to avoid unnecessary precision = 393.665680544729 -3.96284720942939X1 -11.3339537291218X2 + 77.4321045649189X3 erature, heating cost decreases by 3.96284720942939. Moreover, with every one unit increase in insulation, h on line. rved values of the dependent variable (Y) deviate from the predicted values (fitted values from the regression able can be accounted for by the independent variable. n in Y is explained by X measures. This is a very strong strong relationship as it is close to 100% elationship between dependent and independent variables

45649189X3 e held constant. Thus, there is a negative relationship between X1 and Y eld constant. Thus, there is a negative relationship between X2 and Y there is not a garage by than our alpha of .05 (0.00001618 < 0.05) we reject the null hypothesis and conclude that the relationship be than our alpha of .05 (0.0120 < 0.05) we reject the null hypothesis and conclude that the relationship betwee than our alpha of .05 (0.0037 < 0.05) we reject the null hypothesis and conclude that the relationship betwee

Upper 95.0% 489.064142894 -2.5792752288 -2.8510860844 125.729529352 heating cost decreases by 11.3339537291218. Finally, heating cost increases by 77.4321 when there is a presence o n equation) by approximately 41.6184 units higher or lower

etween X1 and Y is statistically significant en X2 and Y is statistically significant en X3 and Y is statistically significant