Explanation Calculation: Multiple linear regression model: A multiple linear regression model is given as y ^ = a + b 1 x 1 + b 2 x 2 + b 3 x 3 + ... + b k x k where y is the response or dependent variable, and x 1 , x 2 , ... , x k are the k quantitative independent variables where k is a positive integer. Here, a is the intercept term of the regression model, that is, the value of predicted value of y when X ’s are 0 and b i ’s are the slopes, that is, the amount of change of the predicted value of y for one unit increase in x i when all other independent variables are constant. In the given problem the dependent variable y is the cost of delivery. The “Prep”, “Delivery”, and “Mileage”, are denoted as x 1 , x 2 , and x 3 , respectively. Step by step procedure to obtain the regression equation using MINITAB software: Choose Stat > Regression > Regression > Fit Regression Model . Under Responses , enter the column of Cost . Under Continuous predictors , enter the columns of Prep , Delivery , and Mileage . Click OK . Output using MINITAB software is given below: Hence, the multiple linear regression equation is y ^ = 14.015 − 0.0703 x 1 − 0.0178 x 2 + 0.903 x 3 . Consider that y is dependent variable and x i ' s are the independent variables where β i ' s are the corresponding population regression coefficient for all i = 1 , 2 , 3 . State the hypotheses: Null hypothesis: H 0 : β 1 = β 2 = β 3 = 0 . That is, the model is not significant. Alternative hypothesis: H 1 : At least one β i is not equal to 0 . That is, the model is significant. In case of global test the F test statistic is defined as, F = SSR k SSE n − k − 1 , where SSR, SSE, n and k are the regression sum of square, error sum of square, sample size and the number of independent variables. According to the output in the value of F statistic is 71.85 with numerator degrees of freedom 3 and denominator degrees of freedom 46. Consider, the level of significance is α = 0.05 . Decision rule: If p -value ≤ α , then reject the null hypothesis. Otherwise failed to reject the null hypothesis. Conclusion: Here, p -value corresponding to the global test is 0. Hence, p -value ( = 0 ) < α ( = 0.05 ) . That is, the p -value is less than the level of significance. Therefore, reject the null hypothesis. Hence, it can be concluded that any of the regression coefficient differ from 0 at 0.05 significance level. For independent variable x 1 : Consider that β 1 is the population regression coefficient of independent variable x 1 . State the hypotheses: Null hypothesis: H 0 : β 1 = 0 . That is, there is no significant relationship between y and x 1 . Alternative hypothesis: H 1 : β 1 ≠ 0 . That is, there is significant relationship between y and x 1 . In case of individual regression coefficient test the t test statistic is defined as, t = b i s b i , where b i and s b i are the i th regression coefficient and the standard deviation of the i th regression coefficient

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