. Estimate the values of β₀ andβ₁

Y₁= β₀+ β₁X₁+u₁

Where Y₁ is the dependent variable

X is the independent variable

β₀ is the y intercept (the value of Y when X is zero)

β₁ is the slope of the line

u₁ is the residual or error term

β₁= N (∑XY)-(∑X) (∑Y)/N (∑X²)- (∑X)²

β₁= 10(4485) -(155) (278)/10(2453) -(155)²

β₁ = 44850-43090/ 24530-24025

β₁ = 1760/505

β₁ = 3.49

β₀ = M_y – β₁M_x

β₀ = 27.8- (15.5×3.49)

β₀ = 27.8-54.095

β₀ = -26.30

2. Calculate the error u₁ for all the observations made

Y₁ = -26.30+ 3.49X₁+u₁

1st observation

Replace the y and x values in the equation

22 = -26.30+ (3.49×12) + u₁

22 = 15.58+ u₁

U₁ = 6.42

2^nd Observation

38 = -26.30+(3.49×17) + u₁

38 = 33.03+u₁

U₁ = 4.97

3^rd Observation

31 = -26.30+ (3.49×16) + u₁

31 = 29.54+ u₁

U₁= 1.46

4^th Observation

15 = -26.30 + (3.49x 13) + u₁

U₁ = -4.07

5^th Observation

20 = -26.30+ (3.49×16) + u₁

U₁ = -9.54

6^th Observation

27 = -26.30+ (3.49×17) + u₁

U₁ = -6.03

7^th Observation

41 = -26.30+ (3.49×19) + u₁

U₁ = 0.99

8^th Observation

22 = -26.30+ (3.49×14) + u₁

U₁ = -0.56

9^th Observation

43 = -26.30+ (3.49×18) + u₁

U₁ = 6.48

10^th Observation

19 = -26.30+ (3.49×13) + u₁

U₁ = -0.07

Summary of the error values

3. Calculate and interpret the standard error of regression

The formula for calculating the regression error is Standard Error = √Sum of error squared/number of observations

The error values give the distance between the actual data points and the regression line. The regression error evaluates the precise nature of the model in predictions based on the dependent variable units. A smaller value of the standard error signifies a lower distance between the regression line and the data points and is therefore desired. A smaller value indicates that the model is sufficiently precise.

The standard error of regression for the sample is calculated below

Standard Error = 5.05

The value obtained is relatively high, thus it can be concluded that the model is not highly sufficient in making the predictions.

4. Calculate and interpret the standard error for B₁

Standard Error = 1.10

5. Test the null hypothesis H₀:B₁=0 (p=0.05)

Test statistics = x-µ/SD/√n

SD = 9.43

Test Statistics = (27.8-0)/(9.43/3.16)

Test Statistics = 9.32

The z score falls outside the range, justifying the rejection of the null hypothesis.

Therefore, there is no sufficient evidence to accept the hull hypothesis