Question

Use the following linear regression equation to answer the questions. x3 = −16.2 + 3.8x1 + 9.6x4 − 1.2x7 (a) Which variable is the response variable? x7 x1 x3 x4 Which variables are the explanatory variables? (Select all that apply.) x4 x7 x1 x3 (b) Which number is the constant term? List the coefficients with their corresponding explanatory variables. constant x1 coefficient x4 coefficient x7 coefficient (c) If x1 = 3, x4 = -1, and x7 = 1, what is the predicted value for x3? (Round your answer to one decimal place.) x3 = (d) Explain how each coefficient can be thought of as a "slope" under certain conditions. If we hold all explanatory variables as fixed constants, the intercept can be thought of 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 other explanatory variables as fixed constants, then we can look at one coefficient as a "slope." Suppose x1 and x7 were held at fixed but arbitrary values. If x4 increased by 1 unit, what would we expect the corresponding change in x3 to be? If x4 increased by 3 units, what would be the corresponding expected change in x3? If x4 decreased by 2 units, what would we expect for the corresponding change in x3? (e) Suppose that n = 16 data points were used to construct the given regression equation and that the standard error for the coefficient of x4 is 0.850. Construct a 90% confidence interval for the coefficient of x4. (Round your answers to two decimal places.) lower limit upper limit (f) Using the information of part (e) and level of significance 1%, test the claim that the coefficient of x4 is different from zero. (Round your answers to two decimal places.) t = t critical = ± Conclusion Fail to reject the null hypothesis, there is insufficient evidence that β4 differs from 0. Fail to reject the null hypothesis, there is sufficient evidence that β4 differs from 0. Reject the null hypothesis, there is sufficient evidence that β4 differs from 0. Reject the null hypothesis, there is insufficient evidence that β4 differs from 0. Explain how the conclusion has a bearing on the regression equation. If we conclude that β4 is not different from 0 then we would remove x4 from the model. If we conclude that β4 is not different from 0 then we would remove x1 from the model. If we conclude that β4 is not different from 0 then we would remove x7 from the model. If we conclude that β4 is not different from 0 then we would remove x3 from the model.

Answer 1

SOLUTION

(a)

From the regression equation, x3 variable is the response variable.

From the regression equation, x1, x4 and x7 are the explanatory variables.

(b)

The constant term and coefficients are

Constant = -16.2

Coefficient of x1 = 3.8

Coefficient of x4 =9.6

Coefficient of x7 = -1.2

(c)

If x1 = 3, x4 = -1, and x7 = 1 the predicted value for x3 is,

x3 = -16.2 + 3.8 * 3 + 9.6 * (-1) - 1.2* 1

= -15.6

(d)

If we hold all other explanatory variables as fixed constants, then we can look at one coefficient as a "slope."

Suppose x1 and x7 were held at fixed but arbitrary values.
If x4 increased by 1 unit, the corresponding change in x3 to be slope of x4 is 9.6

If x4 increased by 3 units, the corresponding expected change in x3 is 3 * 9.6 = 28.8

If x4 decreased by 2 units, the corresponding change in x3 is -2 * 9.6 = -19.2

(e)

Standard error for the coefficient of x4, s = 0.850

Degree of freedom = n-1 = 16 - 1 = 15

t statistic for 90% confidence interval is 1.73

90% confidence interval is

Lower limit = 9.6 - 1.76 * 0.850 = 8.104

Upper limit = 9.6 + 1.76 * 0.850 = 11.096

(f)

t = Coefficient / s = 9.6 / 0.850 = 11.294

t critical for level of significance 5% and df = 19 is 2.144

As, t value is greater than the t critical, we Reject the null hypothesis, there is sufficient evidence that x4 differs from 0.