Question

Use the following linear regression equation to answer the questions.

x₁ = 1.1 + 3.0x₂ – 8.4x₃ + 2.3x₄

Suppose x₃ and x₄ were held at fixed but arbitrary values and x₂ increased by 1 unit. What would be the corresponding change in x₁?


Suppose x₂ increased by 2 units. What would be the expected change in x₁?


Suppose x₂ decreased by 4 units. What would be the expected change in x₁?


(e) Suppose that n = 13 data points were used to construct the given regression equation and that the standard error for the coefficient of x₂ is 0.391. Construct a 95% confidence interval for the coefficient of x₂. (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 x₂ 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 x₃ from the model.

If we conclude that ?₂ is not different from 0 then we would remove x₄ from the model.   

  If we conclude that ?₂ is not different from 0 then we would remove x₁ from the model.

If we conclude that ?₂ is not different from 0 then we would remove x₂ from the model.

Answer 1

Use the following linear regression equation to answer the questions.

x1 = 1.1 + 3.0x2 – 8.4x3 + 2.3x4

Question

Suppose x3 and x4 were held at fixed but arbitrary values and x2 increased by 1 unit. What would be the corresponding change in x1?

There would be an increase of 3.0*1 = 3.0 units in X1 if x3 and x4 were held at fixed but arbitrary values and x2 increased by 1 unit.

Question

Suppose x2 increased by 2 units. What would be the expected change in x1?

There would be an increase of 3.0*2 = 6.0 units in X1 if x3 and x4 were held at fixed but arbitrary values and x2 increased by 2 units.

Question

Suppose x2 decreased by 4 units. What would be the expected change in x1?

There would be an increase of 3.0*4 = 12.0 units in X1 if x3 and x4 were held at fixed but arbitrary values and x2 increased by 4 units.

Part e

We are given

Confidence level = 95%

Standard error = 0.391

Sample size = n = 13

df = n – 2 = 13 – 2 = 11

? = 3.0

Critical t value = 2.2010 (by using t-table/excel)

Margin of error = E = critical t value * standard error

Margin of error = E = 2.2010*0.391 = 0.860591

Confidence interval = ? ± E

Lower limit = ? - E = 3.0 - 0.860591 = 2.139409

Upper limit = ? + E = 3.0 + 0.860591 =3.860591

Lower limit = 2.14

Upper limit = 3.86

Part f

We are given

? = 10% = 0.10

We have to test

H₀: ? = 0 vs. H_a: ? ? 0

Test statistic = t = ?/SE

t = 3.0/0.391 = 7.672634

t = 7.67

t critical ± = 2.2010 (by using t-table/excel)

t critical ± = 2.20

t > t critical

So, we reject the null hypothesis

Reject the null hypothesis, there is sufficient evidence that coefficient of X2 differs from 0.

If we conclude that coefficient of X2 is not different from 0 then we would remove x2 from the model.