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Paragraph AaßbCcDdEe AaBbCcDdEe AaBbCcD A NormalNo Spacing Heading 1 Final Exam Preparation Interpret the three multiple regression coefficients, se, t value, p value, and R. Set up the null and alternative hypotheses for testing about individual regression coefficients and explain the meaning of each and why we need them. (1) (2) (3) Discuss the necessary assumptions for hypothesis testing to reject or not reject the null (4) Conduct the hypothesis testing about individual partial regression coefficients using the test (5) Conduct the confidence interval approach to hypothesis testing. hypothesis of significance approach. -critical t value s calculated I value s +critical-t value)-095 P-2sts +2)-0.95 +2-0.95 .jx) o/x (6) Set up a joint hypothesis and test it. ANOVA Table: df MSSISS/df Source of variation Due to regression (ESS) Due to residual (RSS) Total (TSS uares ESS/ df RSS idf -variance-explained-by-X :-and-X : un explained varianceExplanation / Answer
1.Let us regress Y over X1 and X2.
Then we have 3 regression coefficients beta0, beta1 and beta2 for intercept, X1 and X2 respectively.
Beta0 indicates the value of Y when X1 and X2 are both set to 0.
Beta1 (or beta2) indicates the change in the value of Y when the value of X1 (or X2) is increased by 1 unit keeping the other variable fixed.
se indicates the standard error which is an estimate of the standard deviation of the regression coefficient, the amount by which the coefficient varies across cases.
t-value is the value of the t-statistic obtained by dividing the coefficient by its standard error.
p-value for beta1 (or beta2) is the probability that the t-statistic beta1 (or beta2) will exceed the observed t-value when the null hypothesis of zero beta1 (or beta2) is assumed to be true.
2.Null hypothesis - > Hi0 : beta i = 0, i = 0,1,2 => No significance of Xi (X0 = intercept)
Alternative hypothesis -> Hi1 : beta i 0 , i = 0,1,2 => Significance of Xi
The hypothesis testing is done to check the significance of the intercept, X1 or X2.
3. Assumptions : a. Normality assumption since t-statistic is calculated based on this assumption
b. Independence of errors
c. Linearity
d. Homoschedasticity of errors
e. No multicollinearity ( no correlations between X1 and X2)
4. Here, the t-statistic = Coefficient of beta/standard error of beta.
Critical value = t ; n-2 = Upper 100% point of t-distribution with (n-2) degrees of freedom
If t-statistic > critical value, we reject H0 and conclude that beta is significant else beta is not significant.
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