Answer all the questions:

All questions pertain to the simple (two-variable) linear regression model for which the population regression equation can be written in conventional notation as: Y; =Bo + B,X; + u; (1) where Y, and X; are observable variables, Bo and B, are unknown (constant) regression coefficients, and u; is an unobservable random error term. The Ordinary Least Squares (OLS) sample regression equation corresponding to regression equation (1) is Y, = Bo + B,X; + u; (i = 1, .., N) (2) where Bo is the OLS estimator of the intercept coefficient Bo, B, is the OLS estimator of the slope coefficient B, , u, is the OLS residual for the i-th sample observation, and N is sample size (the number of observations in the sample). (15 marks) 1. Show that the OLS slope coefficient estimator B, is a linear function of the Y, sample values. Stating explicitly all required assumptions, prove that the OLS slope coefficient estimator B, is an unbiased estimator of the slope coefficient B, -2. Give a general definition of the F-distribution. Starting from this definition, derive the F- statistic for the OLS slope coefficient estimator , . State all assumptions required for the derivation.3. A researcher is using data for a sample of 3240 female employees 25 years of age and over to investigate the relationship between employees' hourly wage rates Y; (measured in dollars per hour) and their age Xi (measured in years). The population regression equation takes the form of equation (1): Y; =Bo + B, X; + u; . Preliminary analysis of the sample data produces the following sample information: N N N N = 3240 y= = 78434.97 Ex. = 25526.17 Ex;y; = 3666.426 i=1 i=1 N N EY, = 34379.16 EX; = 96143.00 SY- = 443227.1 i=1 N N N EX- = 2878451.0 2X;Y, = 1023825.0 Euf = 77908.35 i=1 where x; = X; - X and y; = Y, - Y for i = 1, .., N. Use the above sample information to answer all the following questions. Show explicitly all formulas and calculations.(a) Use the above information to compute OLS estimates of the intercept coefficient Bo and the slope coefficient B, .(b) Interpret the slope coefficient estimate you calculated in part (a) - i.e., explain in words what the numeric value you calculated for B, means. Note: B, = 0.1436. Y; is measured in dollars per hour, and X; is measured in years.\f(d) Compute the value of R' , the coefficient of determination for the estimated OLS sample regression equation. Briefly explain what the value you have calculated for R means.(e) Calculate the sample value of the t-statistic for testing the null hypothesis Ho: B, =0 against the alternative hypothesis H1: B, #0. (Note: You are not required to obtain or state the inference of this test.)