## Simple Linear Regression Case Solution

**The three model assumptions for a simple linear regression model.**

1. The relationship between the independent and dependent variable should be linear. The outliers should also check as linear regression model is much sensitive to outliers.

2. All variables of linear regression analysis should be multivariate normal. It can be checked with a goodness of fit test.

3. The assumption we will take is that thereshould be no multicollinearity in the data. This assumption is taken on the basis of relationship between independent variable and dependent variable. The independent variables are not independent from one another.

The assumptions for linear regression model should be checked asit can subject the validity of the linear regression model. The assumptions are checked by plotting three different plots.

**(i) Normal probability plot of residuals**

The normal probability plot of residuals helps us to evaluate our second assumption,in which we examine that multicollinearity exists in the data. It can be seen through the probability plot of residuals that all data is normally distributed through linear alteration. The plot shows that errors are not increasing over time. The data has the capability to produce quality of confidence intervals.

**(ii) Plot of residual vs. Predicted values**

The plot of residuals as compared to predicted values evaluates the third model assumption for the linear regression model. The purpose of this assumption is completing the data accomplished by this plot. It can be seen on the plot that the points are not scattered and not overlapping each other. Only few outliers can be seen in this plot, which shows that the third assumption for the model is also verified. The interdependency of variables can also be further be identified by correlation matrix.

**(iii) Normal probability plot of house price**

The first assumption of the model can be evaluated by this plot. The reason of evaluating this assumption is that if the dependent and independent variables are not linear, then the output of the data will be false and based on superstitious predictions. The dependent variable taken for this assumption is house price. It can be seen on the plot that there are few outliers in the data and the points are also systemically attached to each other. It shows that the first assumption for the model is accepted.

**Scatter plot matrix**

The purpose of the scatter plot matrix is to show the sketch of scatter plots of variables with other variables. It suggests if there is a need of transformations for the variables to take place. The scatter plot matrix for this case is to examine linearity between variables.

In this scatter plot matrix, it can be seen that thecolumns for the price of house and size of house createa linear relationship with each other. The column for price in evaluating linearity with bedroom and bathroom are not very clear. In case ifscatter plot results are not clear, then we can move to columns of variables; number of bedrooms and number of bathrooms in comparison with price of the house.

A non-linearity is observed by viewing the columns of size of house variable in contrast with other two variables. The columns for the number of bedrooms and the number of bathrooms are also not very clear. A non-linearity between the variables can be observed, which is due to non-interdependency of variables on each other.

**Correlation matrix**

The correlation matrix is used to evaluate the significance of correlation of variables and linearity of data. A strong Pearson correlation of .899 between the variables; price of house and size of house can be observed. A Pearson correlation of .590 and .714 of price of house with number of bedrooms and bathrooms is observed, which shows that the correlation of price with other variables is also good...............

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