Use the following linear regression equation to answer the questions.
x_{1} = 1.1 + 3.0x_{2} –
8.4x_{3} + 2.3x_{4}
(a) Which variable is the response variable?
x_{3}
x_{1}
x_{2}
x_{4}
Which variables are the explanatory variables? (Select all that
apply.)
x_{1}
x_{2}
x_{3}
x_{4}
(b) Which number is the constant term? List the coefficients with
their corresponding explanatory variables.
constant = | |
x_{2} coefficient= | |
x_{3} coefficient= | |
x_{4} coefficient= |
(c) If x_{2} = 4, x_{3} = 10, and
x_{4} = 6, what is the predicted value for
x_{1}? (Use 1 decimal place.)
(d) Explain how each coefficient can be thought of as a "slope"
under certain conditions.
If we hold all other explanatory variables as fixed constants, then we can look at one coefficient as a "slope."
If we look at all coefficients together, each one can be thought of as a "slope."
If we look at all coefficients together, the sum of them can be thought of as the overall "slope" of the regression line.
If we hold all explanatory variables as fixed constants, the intercept can be thought of as a "slope."
Suppose x_{3} and x_{4} were held
at fixed but arbitrary values and x_{2} increased
by 1 unit. What would be the corresponding change in
x_{1}?
Suppose x_{2} increased by 2 units. What would be
the expected change in x_{1}?
Suppose x_{2} decreased by 4 units. What would be
the expected change in x_{1}?
(e) Suppose that n = 13 data points were used to construct
the given regression equation and that the standard error for the
coefficient of x_{2} is 0.391. Construct a 95%
confidence interval for the coefficient of x_{2}.
(Use 2 decimal places.)
lower limit= | |
upper limit = |
(f) Using the information of part (e) and level of significance
10%, test the claim that the coefficient of x_{2}
is different from zero. (Use 2 decimal places.)
t= | |
t critical ± | = |
Conclusion
Reject the null hypothesis, there is sufficient evidence that ?_{2} differs from 0.
Reject the null hypothesis, there is insufficient evidence that ?_{2} differs from 0.
Fail to reject the null hypothesis, there is insufficient evidence that ?_{2} differs from 0.
Fail to reject the null hypothesis, there is sufficient evidence that ?_{2} differs from 0.
Explain how the conclusion of this test would affect the regression
equation.
If we conclude that ?_{2} is not different from 0 then we would remove x_{3} from the model.If we conclude that ?_{2} is not different from 0 then we would remove x_{4} from the model.
If we conclude that ?_{2} is not different from 0 then we would remove x_{1} from the model.If we conclude that ?_{2} is not different from 0 then we would remove x_{2} from the model.
Answer :
A) response variable = x1, it is the variable where the value calculated as a response from input is stored.
Explanatory variable : x2,x3 and x4. These are independent variables system is having.
B) constant term : 1.1
Coefficient of x2 = 3.0
Coefficient of x3 = -8.4
Coefficient of x4= 2.3
C) given x2 = 4, x3 = 10 and x4 = 6: then using given equation
X1 = 1.1 + 3.0*4 - 8.4*10+2.3* 6. = -57.1
D) If we hold all other explanatory variables as fixed constants, then we can look at one coefficient as a "slope."
Example : if x3 and x4 are zero that is de =0 and x4=0 than the coefficient of x2 will be the slope of the line and this is valid in all the other cases also.
If x3 and x4 are constant and de is increase by 1, then the x1 will get increase by the factor equals to coefficient of x2.
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