다중선형회귀 (Multiple linear regression)

Sales = $\beta_0$ + $\beta_1$youtube + $\beta_2$facebook + $\beta_3$newspaper + $\epsilon_i$

Q1. Write the assumptions we impose on $\epsilon$
>> $\epsilon$ ~ iid N(0, $\sigma^2$)


Q2. Using the OLS method, estimate regression coefficients.
$\hat{\beta_0}$ = 3.52667
$\hat{\beta_1}$ = 0.04576
$\hat{\beta_2}$ = 0.18853
$\hat{\beta_3}$ = -0.00104


Q3. Estimate $\sigma^2 $
4.09096

없으면 root MSE 제곱해서 답 적기($2.02261 ^ 2$)
SSE / DF = MSE


Q4. Test the overall utility of the model.
$H_0: \beta_1 = \beta_2 = \beta_3 = 0$
$H_1: \beta_j \neq 0$ for some j = 1,2,3

Let $\alpha = 0.05$
Since p-value =< 0.0001 is less than $\alpha = 0.05$, we reject $H_0$.
Thus, the overall model is useful.


Q5. Can we claim that the facebook advertising is related to sales? Justify.
$H_0: \beta_2 = 0$
$H_1: \beta_2 \neq 0$

Let $\alpha$ = 0.05.
Since p-value =< 0.0001 is less than $\alpha$ = 0.05, we reject $H_0$.
That is, we can claim that the facebook advertising is related to sales.


Q5-1. Can we claim that the newspaper advertising is related to sales? Justify.
$H_0$: $\beta_3 = 0$
$H_1$: $\beta_3 \neq 0$

Let $\alpha$ = 0.05.
Since p-value = 0.8599 is greater than $\alpha$ = 0.05, we fail to reject $H_0$.
That is, we can claim that the facebook advertising is related to sales.
This means that the newspaper advertising is not related to sales.


Q6. Compute the expected change in the sales when our spending for youtube advertisement increases by 1000달러 while the others are fixed. 
45.76달러(0.04576 * 1000달러)


Q7. Compute the expected sale when we spend 1000달러 for each method of adv.
$\hat{Sale}$
45.76 + 188.53 + 3526.67 - 1.04= 3.7599 * 1000

Q7-1. Compute the expected sale when we spend 0달러 for all of adv.
x들의 값이 0일 때의 기댓값이 위의 질문에 대한 대답
$\beta_0$ => 3.52667 * 1000달러

$\beta_0$는 실제의 데이터 변수의 성질에 따라 x들의 값을 0으로 둘 수 있는지 없는지가 갈린다. 

Q8. Can we argue that youtube is negatively related to the sale?
1. youtube랑 sales와 관계가 있는지부터 확인($\beta$)
2. 그러고 난 다음에 관계가 있으면 $\hat{\beta}$의 값을 신뢰할 수 있음. 
   왜냐하면 $\beta_1$의 값이 0이 아니라고 1번에서 결론지었기 때문에

We conclude that $\beta_1 \neq 0$.
However, since $\hat{\beta_1}$ = 0.04576 is greater than 0, we cannot argue that...

Q8-1. Can we argue that newspaper is negatively related to the sale?
We conclude that $\beta_3$ = 0. We cannot argue that ... 


Q9. Compute the R-squared and interpret it.
$R^2$ = 0.8972

89.72% variability in sales can be explained 
by the fitted model (with youtube, facebook, and newspaper.)


Q9-1. adj. R-squared 도 마찬가지.