16. By what criteria is an ordinary (simple) linear regression considered \"best

Question

16. By what criteria is an ordinary (simple) linear regression considered "best fit?" O A. A straight line is drawn between all of the points o B. The slope is calculated from an equation that maximizes the sums of squares for the regression a C. The slope is calculated from an equation that minimizes the sums of squares for the regression D. The slope is calculated frorn an equation that maximizes the sums of squares for the error (residuals) E. The slope is calculated from, an equation that minimizes the sums of squares for the error (residuals) 17. Which is the single biggest limitation of multiple linear regression? o A. If the independent variables are not linearly related to the dependent variable, the regression slopes cannot be B. If the dependent variable is not correlated with any of the independent variables, the regression slopes cannot be C. If the independent variables are not correlated with each other, the regression slopes cannot be tested and D. If the independent variables are correlated with each other, the regression slopes cannot be tested and E. None of the above tested and interpreted. tested and interpreted. interpreted interpreted. 18. How many degrees of freedom are there for a test of significance for the regression of Y on X using the ANOVA F-test method? A. F numerator df= l and denominator df= n-1 B. F numerator df-1 and denominator df= n-2 C. F numertor df= n-1 and denominator df-1 D. F numertor df= n-2 and denominator df= I E. None of the above

Explanation / Answer

16. ordinary linear regression is formulated using the least square method.

let the regression model is

Y=a+bX+e where a and b are the regression parameters which are estimated using least square method.

and in least square method the sum of squares of errors is minimized with respect to a and b

because the idea behind linear regression is to draw a straight line that minimizes the errors

hence the correct option is option E: the slope is calculated from an equation that minimizes the sum of squares of error (residual)

17. the single biggest limitation of multiple linear regression is the problem of multicollinearity.

where the independent variables are related to each other, then the regression cofficients become faulty and fails to interpret the model successfully. for example let the regression model be

Y=a+bX1+cX2

where interpretation of b is , b is the amount of change in Y for unit change in X1 keeping X2 fixed. but if X1 and X2 are correlated then changing X1 would also change X2

hence the fact "keeping X2 fixed" is violated. therefore the interpretation of b becomes faulty.

henc correct option is option D. if the independent variables are correlated with each other, the regression slopes can not be tested and interpreted.

18. the regression model is Y=a+bX

to test the significance of the equation null hypothesis is H0:b=0 and alternative hypothesis is H1: not H0

let number of observations is n

then total sum of squares would have n-1 degrees of freedom

and sum of squares due to regression would have 1 degrees of freedom as there is only one parameter b under testing

and since df of total sum of squares=df of sum of squares due to regression+df of sum of squares due to error

df of sum of squares due to error=n-1-1=n-2

and for F test,the numerator df=df of sum of squares due to regression

denominator df =df of sum of squares due to error

so correct option is option B. F numerator df=1 and denominator df=n-2

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