In the Multiple Regression Model with Two Explanatory Variables $$Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { 1 i } + \beta _ { 2 } X _ { 2 i } + u _ { i }$$

question 46

Essay

In the multiple regression model with two explanatory variables $$Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { 1 i } + \beta _ { 2 } X _ { 2 i } + u _ { i }$$ the OLS estimators for the three parameters are as follows (small letters refer to deviations from means as in $z _ { i } = Z _ { i } - \bar { Z }$ ):
$$\begin{array} { c } \hat { \beta } _ { 0 } = \bar { Y } - \hat { \beta } _ { 1 } \bar { X } _ { 1 } - \hat { \beta } _ { 2 } \bar { X } _ { 2 } \\\\\hat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } } { \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } \right) ^ { 2 } } \\\\\hat { \beta } _ { 2 } = \frac { \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } - \sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } } { \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } \right) ^ { 2 } }\end{array}$$
You have collected data for 104 countries of the world from the Penn World Tables and want to estimate the effect of the population growth rate $\left( X _ { 1 i } \right)$ and the saving rate $\left( X _ { 2 i } \right)$ (average investment share of GDP from 1980 to 1990 ) on GDP per worker (relative to the U.S.)in 1990.The various sums needed to calculate the OLS estimates are given
below: $$\begin{array} { c } \sum _ { i = 1 } ^ { n } Y _ { i } = 33.33 ; \sum _ { i = 1 } ^ { n } X _ { 1 i } = 2.025 ; \sum _ { i = 1 } ^ { n } X _ { 2 i } = 17.313 \\\\\sum _ { i = 1 } ^ { n } y _ { i } ^ { 2 } = 8.3103 ; \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } = .0122 ; \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } = 0.6422 \\\\\sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } = - 0.2304 ; \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } = 1.5676 ; \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } = - 0.0520\end{array}$$ The heteroskedasticity-robust standard errors of the two slope coefficients are 1.99 (for
population growth)and 0.23 (for the saving rate).Calculate the 95% confidence interval
for both coefficients.How many standard deviations are the coefficients away from zero?

Calculate centerline and control limits for various production processes.

Identify and explain the use of pooled standard deviation in process control.

Plot sample means and understand the significance of sample size in process monitoring.

Distinguish between charts used for monitoring process mean and variability.