# Multiple Regression Analysis Detailed Example: Delivery Time Multiple Regression

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**Analysis Step 1**

At the first step, the following model is fitted the data to explain the relationship between the delivery time (y) and the number of cases delivered (x1) and the distance between the center and the delivered place (x2).

(See the APPENDIX 1 for the MINITAB output)

The fitted regression equation is obtained as follows:

Delivery Time (y)= 2,34 + 1,62 Number of Cases (x1) + 0,0144 Distance, x2 (ft).

To be sure that the validity of the model fitted, we check the LINE assumptions. For this purpose, the following residual analysis is conducted:

**Check for LINE assumptions related to the above fitted model:**

**1.** **Normality of errors:**

When we look at the Normal Probability Plot (NPP) of (standardized) residuals, it can be seen that standardized residuals do not follow the reference line closely; also there seems to be an outlier observation, whose standardized residual is greater than 3.

To be sure that there is or not normality problem with the errors, we should conduct the most powerful normality test, called Shapiro-Wilk’s test. Note here that in MINITAB, we select Ryan-Joiner (RJ)which is similar to Shapiro-Wilk’s test. Following are the null and the alternative hypotheses for conducting the test.

H0: Errors are distributed normally

versus (vs.)

H1: Errors are not distributed normally

Since the p-value associated with the RJ test given in the NPP of standardized residuals is 0.044, which is less than the default Type-I error probability, α=0.05, we do reject H0, which states normality of errors. Hence, normality assumption of errors is failed; that is, errors do not follow normal distribution.

**2.** **Constant error variance:**

When we examine standardized residuals versus fitted values graphics above, there does not seem to be an error variance heterogeneity problem…