From the linear model we find expected u

Using this formula we find out the t_stat

Our t_crit is 2.052 (df is 27, α is 0.05)

t_stat value is greater than t_crit, which means that we reject the H₀ and conclude that the sum of the coefficients of X_1,i and X_2,i is not negative 78.

2. 
   Partial coefficient of correlation is obtained due to the reason that we need a coefficient of correlation that is free of influence, if any, of X₃ on X₂ and Y. Also, it can be defined as following:
   

r_{1 2.3} = PCC between Y and X₂, holding X₃ constant

r_{1 3.2} = PCC between Y and X₃, holding X₂ constant

r_{2 3.1} = PCC between X₂ and X₃, holding Y constant

Question 4

1. 
   
   

2. 
   NGDP_t= -986.3317 +201.9772T_t + u_t
   

SE= 212.6816 9.267491 R² = 0.9277

t= -4.64 21.79

RGDP_t= 1907.715 +128.782T_t + u_t

SE= 45.13298 1.966646 R² = 0.9914

t= 42.27 65.48

3. 
   On average, the nominal GDP of USA has increased 202 USD annually starting from 1959 year.
   

On average, the real GDP of USA has increased 129 USD annually starting from 1959 year.

4. 
   There is a difference between the coefficients of current and constant dollar GDP. The difference is on average is about 73 USD. This difference can be explained by the inflation that is included in RGDP.
   
5. 
   The graph and regression analysis shows that, nominal GDP is increasing in a faster pace compared to the real one. This can be explained by the fact that RGDP is adjusted to the inflation and the rate of inflation in USA is increasing.
   

Question 5

1. 
   According to the regression analysis, the linear model has been constructed:
   

Pay_i = 12129.52 + 3.307597Spend_i + u_i

SE= 1197.408 0.3117191 R²=0.6968

RSS=264850534 SSE=608559605

2. 
   H_o: β₂=3.0
   

H_a: β₂≠3.0

β₂=3.307597± 2*(0.3117191) => β₂= [2.68411588; 3.9310352] => this is the 95% confidence interval

the interval shows that the coefficient of a slope is inside this interval with a 95% confidence.

3. 
   Individual and mean forecast values when spending is 5000$ is Pay_i = 12129.52 + 3.307597*5000 =
   

=28679.63452

Standard error of the mean value is 520.5117, which is found by the formula

Standard error of the individual value is 2382.337, which is found by the formula

Then the confidence intervals for mean individual predictions are calculated: