Answer:
Step-by-step explanation:
Hello!
Given the data of the multiple linear regression (see attachment) You have to conduct the individual tests for the five independent variables proposed to determine which one is significantly relevant for the regression.
The individual hypothesis tests of the multiple regression are the same as conducting the analysis for simple linear regression. The parameter of interest will be the population slope of the given variable, βi, and the statistic will be the Student's t: [tex]t= \frac{b_i-\beta_i }{Sb} ~ t_{n-2}[/tex]
All hypothesis tests will be two tailed, so using α: 0.05, the critical region and decision rule for the five tests will be:
[tex]t_{n-2;\alpha /2}= t_{23;0.025}= -2.069[/tex]
[tex]t_{n-2;1-\alpha /2}= t_{23;0.975}= 2.069[/tex]
If [tex]t_{H_0}[/tex] ≤ -2.069 or if [tex]t_{H_0}[/tex] ≥ 2.069, then the decision is to reject the null hypothesis.
If -2.060<[tex]t_{H_0}[/tex] <2.069, then the decision is to not reject the null hypothesis.
Individual tests:
H₀: β₁=0
H₁: β₁≠0
α:0.05
b₁= 4.00
Sb₁= 1.50
[tex]t_{H_0}[/tex]= 2.00
Since -2.060<[tex]t_{H_0}[/tex] <2.069, the decision is to not reject the null hypothesis.
H₀: β₂=0
H₁: β₂≠0
α:0.05
b₂= 3.00
Sb₂= 0.20
[tex]t_{H_0}[/tex]= 15.00
Since [tex]t_{H_0}[/tex] ≥ 2.069, the decision is to reject the null hypothesis.
H₀: β₃=0
H₁: β₃≠0
α:0.05
b₃= 0.20
Sb₃= 0.05
[tex]t_{H_0}[/tex]= 4.00
Since [tex]t_{H_0}[/tex] ≥ 2.069, the decision is to reject the null hypothesis.
H₀: β₄=0
H₁: β₄≠0
α:0.05
b₄= –2.50
Sb₄= 1.00
[tex]t_{H_0}[/tex]= –2.50
Since [tex]t_{H_0}[/tex] ≤ -2.069, the decision is to reject the null hypothesis.
H₀: β₅=0
H₁: β₅≠0
α:0.05
b₅= 3.00
Sb₅= 4.00
[tex]t_{H_0}[/tex]= 0.75
Since -2.060<[tex]t_{H_0}[/tex] <2.069, the decision is to not reject the null hypothesis.
Even though it seems that the five independent variables altogether conform to a good regression model to explain/ predict the dependent variable. Individually there is not significant evidence to conclude that X₁ and X₅ are good predictors of Y. Therefore the model could work just as well using only X₂, X₃, X₄ as explanatory variables or there is a chance that without those two variables the regression will be even stronger.
I hope it helps!
