Answer:
Step-by-step explanation:
n = sample size = 5
a) Let us determine the sum
![\sum x_i= 100+200+300+400+490=1490](https://tex.z-dn.net/?f=%5Csum%20x_i%3D%20100%2B200%2B300%2B400%2B490%3D1490)
![\sum x_i^2= 100^2+200^2+300^2+400^2+490^2=540100](https://tex.z-dn.net/?f=%5Csum%20x_i%5E2%3D%20100%5E2%2B200%5E2%2B300%5E2%2B400%5E2%2B490%5E2%3D540100)
![\sum y_i = 237+350+419+465+507=1978](https://tex.z-dn.net/?f=%5Csum%20y_i%20%3D%20237%2B350%2B419%2B465%2B507%3D1978)
![\sum y_i^2= 237^2+350^2+419^2+465^2+507^2=827504](https://tex.z-dn.net/?f=%5Csum%20y_i%5E2%3D%20237%5E2%2B350%5E2%2B419%5E2%2B465%5E2%2B507%5E2%3D827504)
![\sum x_i y_i=100 \times 237 + 200\times350+300 \times 419 + 400 \times 465 + 490 \times 507=653830](https://tex.z-dn.net/?f=%5Csum%20x_i%20y_i%3D100%20%20%5Ctimes%20237%20%2B%20200%5Ctimes350%2B300%20%5Ctimes%20419%20%2B%20400%20%5Ctimes%20465%20%2B%20%20490%20%5Ctimes%20507%3D653830)
Now we can determine ![S_x_x, S_x_y, S_y_y](https://tex.z-dn.net/?f=S_x_x%2C%20S_x_y%2C%20S_y_y)
![S_x_x = \sum x_i^2-\frac{(\sum x_1)^2}{n} \\= 540100 - \frac{1490^2}{5} \\= 96080](https://tex.z-dn.net/?f=S_x_x%20%3D%20%5Csum%20x_i%5E2-%5Cfrac%7B%28%5Csum%20x_1%29%5E2%7D%7Bn%7D%20%5C%5C%3D%20540100%20-%20%5Cfrac%7B1490%5E2%7D%7B5%7D%20%5C%5C%3D%2096080)
![S_x_y = \sum x_i y_i -\frac{(\sum x_i)(\sum y_i) }{n} \\\\](https://tex.z-dn.net/?f=S_x_y%20%3D%20%5Csum%20x_i%20y_i%20-%5Cfrac%7B%28%5Csum%20x_i%29%28%5Csum%20y_i%29%20%7D%7Bn%7D%20%5C%5C%5C%5C)
![653830 - \frac{1490 \times 1978 }{5} = 64386](https://tex.z-dn.net/?f=653830%20-%20%5Cfrac%7B1490%20%5Ctimes%201978%20%7D%7B5%7D%20%20%3D%2064386)
![S_y_y = \sum y_i^2-\frac{(\sum y_i)^2 }{n} = 82750-\frac{1978^2}{5} \\\\= 45007.2](https://tex.z-dn.net/?f=S_y_y%20%3D%20%5Csum%20y_i%5E2-%5Cfrac%7B%28%5Csum%20y_i%29%5E2%20%7D%7Bn%7D%20%3D%2082750-%5Cfrac%7B1978%5E2%7D%7B5%7D%20%5C%5C%5C%5C%3D%2045007.2)
The estimate b of the slope β is the ratio of
and ![S_x_x](https://tex.z-dn.net/?f=S_x_x)
![b = \frac{S_x_y}{S_x_x}](https://tex.z-dn.net/?f=b%20%3D%20%5Cfrac%7BS_x_y%7D%7BS_x_x%7D)
![\frac{64386}{96080} = 0.67](https://tex.z-dn.net/?f=%5Cfrac%7B64386%7D%7B96080%7D%20%20%3D%200.67)
The mean is the sum of all value divide by number of values
![\bar x= \frac{\sum x_i}{n} \\\\= \frac{100+200+300+400+490}{5} \\\\= \frac{1490}{5} = 298](https://tex.z-dn.net/?f=%5Cbar%20x%3D%20%5Cfrac%7B%5Csum%20x_i%7D%7Bn%7D%20%5C%5C%5C%5C%3D%20%5Cfrac%7B100%2B200%2B300%2B400%2B490%7D%7B5%7D%20%5C%5C%5C%5C%3D%20%5Cfrac%7B1490%7D%7B5%7D%20%3D%20298)
![\bar y= \frac{\sum y_i}{n} \\\\= \frac{237+350+419+465+507}{5} \\\\= \frac{1978}{5} = 395.6](https://tex.z-dn.net/?f=%5Cbar%20y%3D%20%5Cfrac%7B%5Csum%20y_i%7D%7Bn%7D%20%5C%5C%5C%5C%3D%20%5Cfrac%7B237%2B350%2B419%2B465%2B507%7D%7B5%7D%20%5C%5C%5C%5C%3D%20%5Cfrac%7B1978%7D%7B5%7D%20%3D%20395.6)
The estimate a of the intercept is
![a = \bar y - b \bar x](https://tex.z-dn.net/?f=a%20%3D%20%5Cbar%20y%20-%20b%20%5Cbar%20x)
![= 395.6 - 0.69 \times 298\\= 195.9](https://tex.z-dn.net/?f=%3D%20395.6%20-%200.69%20%5Ctimes%20298%5C%5C%3D%20195.9)
General least square equation;
![\bar y = \alpha + \beta x](https://tex.z-dn.net/?f=%5Cbar%20y%20%3D%20%5Calpha%20%2B%20%5Cbeta%20x)
replace alpha by a = 3 and beta by b = 0.67 in general least equation
y = a + bx
195.9 + 0.67x
b)
<em>Scatter plot is shown in the attached file</em>
x is on the horizontal axis
y is n the vertical axis
The degree of freedom of regression is 1
because we use one variable s predictor variable
![d_f_R = 1](https://tex.z-dn.net/?f=d_f_R%20%3D%201)
The degree of freedom of error is the sample size n decrease by 2
![d_f_E =n-2= 5 - 2=3](https://tex.z-dn.net/?f=d_f_E%20%3Dn-2%3D%205%20-%202%3D3)
Total df is equal to the sum of seperate degree of freedom dfR and dfE
total df = 1 +3 4
![SSR = \frac{(S_x_y)^2}{S_x_x} = \frac{64386^2}{96080} \\\\= 43146.9296](https://tex.z-dn.net/?f=SSR%20%3D%20%5Cfrac%7B%28S_x_y%29%5E2%7D%7BS_x_x%7D%20%3D%20%5Cfrac%7B64386%5E2%7D%7B96080%7D%20%5C%5C%5C%5C%3D%2043146.9296)
Total SS =Syy= 45007.2
SSE + Total SS = SSR
= 45007.2 - 43146.9296
= 1860.2705