ads<-read.csv("Advertising.csv", header=TRUE)
attach(ads)

names(ads)

## [1] "X"         "TV"        "radio"     "newspaper" "sales"

head(ads)

##   X    TV radio newspaper sales
## 1 1 230.1  37.8      69.2  22.1
## 2 2  44.5  39.3      45.1  10.4
## 3 3  17.2  45.9      69.3   9.3
## 4 4 151.5  41.3      58.5  18.5
## 5 5 180.8  10.8      58.4  12.9
## 6 6   8.7  48.9      75.0   7.2

cor(TV, sales)

## [1] 0.7822244

plot(TV, sales, main= "Correlation of Sales by TV")

#use lm(response, explanatory)
mod1<-lm(sales~TV)
mod1

## 
## Call:
## lm(formula = sales ~ TV)
## 
## Coefficients:
## (Intercept)           TV  
##     7.03259      0.04754

#the slope coefficient of approximately 0.05 tells us that for every 1 TV ads run we get a new  .05 sales.

confint(mod1)

##                  2.5 %     97.5 %
## (Intercept) 6.12971927 7.93546783
## TV          0.04223072 0.05284256

#using a F-test for a simple linear regression
#with simple linear regression the t and f tests are indentical
summary(mod1)

## 
## Call:
## lm(formula = sales ~ TV)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -8.3860 -1.9545 -0.1913  2.0671  7.2124 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 7.032594   0.457843   15.36   <2e-16 ***
## TV          0.047537   0.002691   17.67   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.259 on 198 degrees of freedom
## Multiple R-squared:  0.6119, Adjusted R-squared:  0.6099 
## F-statistic: 312.1 on 1 and 198 DF,  p-value: < 2.2e-16

#this gives out an f-statistic similar to the anova, and also gives the t-value
# this is a t-test for slope!!!
anova(mod1)

## Analysis of Variance Table
## 
## Response: sales
##            Df Sum Sq Mean Sq F value    Pr(>F)    
## TV          1 3314.6  3314.6  312.14 < 2.2e-16 ***
## Residuals 198 2102.5    10.6                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

#gives the f-statistic 

#the anova and the summary give the same information becasue the f model and t model are equivalent becasue they both test the same hypothesis. reesults in the same p-value


library(ggplot2)

ggplot(ads, aes(x=TV, y=sales))+
  geom_point()+
  geom_abline(slope=mod1$coefficients[2], intercept=mod1$coefficients[1],
              color="blue", lty=2, lwd=1)+
  theme_dark()

#Much less than .05

#to do prediction we must pass in a new data frame (for one thing as below we must pass through in c() function)
newdata<-data.frame(TV= c(100))

predict(mod1, newdata, 
        interval="predict")

##        fit      lwr      upr
## 1 11.78626 5.339251 18.23326

#PI=11.78626 5.339251 18.23326

predict(mod1, newdata, 
        interval="confidence")

##        fit      lwr     upr
## 1 11.78626 11.26782 12.3047

#CI= fit      lwr     upr
 #11.78626 11.26782 12.3047

#the confidence interval averages the error across a group and has less standard error than just one observation.
#if you think about this then you can see that predicting one value will have a larger amount of possible error, as we cannot rely on the canceling out of error above and below, which occurs when using a mean (center) of data.