For the question, consider the following sample regression line: Salary_1 = 50 +
ID: 3150803 • Letter: F
Question
For the question, consider the following sample regression line: Salary_1 = 50 + 3 educ_1 - 1.2 experience_1 Salary is measured in thousands of dollars and education and experience are both measured by numbers of years Assume that the standard errors for the coefficient estimates are.5 for education and 1.5 for experience. Additionally, assume R^2 =.57 and R^2 =-54. The estimates come from a sample of 12 observations (assume that the error term for the regression is normally distributed). What is the population regression line that has been estimated by the sample regression line? Interpret R^2, R^2, and the coefficient estimates for beta_0, beta_1 and beta_2. Do these estimates make intuitive sense? Conduct a positive, right-tailed hypothesis test for the coefficient estimate on education at the 99% level of confidence. Show all steps and be sure to interpret your results. Conduct a two-tailed hypothesis test for the coefficient estimate on experience at the 95% level of confidence. Show all steps and be sure to interpret your results. Construct a 95% confidence interval for the coefficient on education. If you were to extend t his model to include more independent variables, what would you include? There are seven assumptions required for Ordinary Least Squares (OLS) to be the Best Linear Unbiased Estimator (BLUE). For each assumption, answer the following: What is the assumption? How can the assumption be violated? If it is violated, what do we call the violation? How can you detect if this assumption is violated? What is the consequence of violating this assumption? (Biased estimate of beta? Wrong standard errors? Both?) What is a potential fix if this assumption is not true?Explanation / Answer
Given that the regression line is,
salary yi = 50 + 3*edu - 1.2*experience.
Here we can say that there are two independent variables as education and experience and dependent variable is salary.
Intercept (B0) = 50
Slope coefficient for education (b1) = 3
Slope coefficient for experience (b2) = -1.2
The population regression line is,
Y = B0+ B1x1 + B2x2
where B0 is intercept.
B1 is slope for x1.
and B2 is slope for x2.
R^2 = 0.57
It expresses the proportion of the variation in y which is explained by variation in independent variables.
Interpretation of B0, B1 and B2 :
If experience value is fixed then for each change of 1 unit in education salary increases by 3 units.
Similarly if education value is fixed then for each change of 1 unit in experience salary decreases by 1.2 units.
We are given that standard error values.
Standard error for education = 0.5
Standard error for experience = 1.5
Test statistic t = regression coefficient / standard error
t for education :
t = 3 / 0.5 = 6
t for experience :
t = -1.2 / 1.5 = -0.8
Assume alpha = 0.01
Here test of hypothesis is,
H0 : Be = 0 H1 : Be > 0
where Be is population slope for education.
P-value for education :
EXCEL syntax :
=TDIST(x, deg_freedom, tails)
where x is test statistic value.
deg_freedom = n-1 = 12 - 1 = 11
tails = 1
P-value = 0.00004
P-value < alpha
Reject H0 at 1% level of significance.
Conclusion : Population slope for education is differ than 0.
Here the hypothesis for the test is,
H0 : Bex = 0 Vs H1 : Bex 0
Bex is population slope for experience.
Assume alpha = 5% = 0.05
P-value = 0.44
P-value > alpha
Accept H0 at 5% level of significance.
Conclusion : Population slope for experience is 0.
We can include the independent variable which variable has P-value < alpha.
And that variable is education.
95% confident interval for the coefficient on education is,
b - E < B < b + E
where b is sample regression coefficient for education = 3
E = tc * SE
where tc is the t-critical value for t-distribution.
SE is the standard error for education.
tc we can find by using EXCEL.
syntax :
=TINV(probability, deg_freedom)
where probability = 1-c
c is confidence level = 95% = 0.95
deg_freedom = n-1 = 12 - 2 =10
tc = 2.228
E = 2.228*0.5 = 1.114
lower limit = b - E = 3 - 1.114 = 1.886
upper limit = b + E = 3 + 1.114 = 4.114
95% confidence interval for slope of education is (1.886, 4.114)
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Assumptions :
1) Linearity in parameters alpha and beta: the DV is a linear function of a set of IV and a random error component.
2) The expected value of the error term is zero for all observations.
3) Homoskedasticity
4) Error term is independently distributed and not correlated, no correlation between observations of the DV.
5) Xi is deterministic.
6) Other problems : measurement errors, multicolinearity.
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