d <- airquality
str(d)

'data.frame':   153 obs. of  6 variables:
 $ Ozone  : int  41 36 12 18 NA 28 23 19 8 NA ...
 $ Solar.R: int  190 118 149 313 NA NA 299 99 19 194 ...
 $ Wind   : num  7.4 8 12.6 11.5 14.3 14.9 8.6 13.8 20.1 8.6 ...
 $ Temp   : int  67 72 74 62 56 66 65 59 61 69 ...
 $ Month  : int  5 5 5 5 5 5 5 5 5 5 ...
 $ Day    : int  1 2 3 4 5 6 7 8 9 10 ...

head(d)

  Ozone Solar.R Wind Temp Month Day
1    41     190  7.4   67     5   1
2    36     118  8.0   72     5   2
3    12     149 12.6   74     5   3
4    18     313 11.5   62     5   4
5    NA      NA 14.3   56     5   5
6    28      NA 14.9   66     5   6

summary(d)

     Ozone           Solar.R           Wind             Temp      
 Min.   :  1.00   Min.   :  7.0   Min.   : 1.700   Min.   :56.00  
 1st Qu.: 18.00   1st Qu.:115.8   1st Qu.: 7.400   1st Qu.:72.00  
 Median : 31.50   Median :205.0   Median : 9.700   Median :79.00  
 Mean   : 42.13   Mean   :185.9   Mean   : 9.958   Mean   :77.88  
 3rd Qu.: 63.25   3rd Qu.:258.8   3rd Qu.:11.500   3rd Qu.:85.00  
 Max.   :168.00   Max.   :334.0   Max.   :20.700   Max.   :97.00  
 NA's   :37       NA's   :7                                       
     Month            Day      
 Min.   :5.000   Min.   : 1.0  
 1st Qu.:6.000   1st Qu.: 8.0  
 Median :7.000   Median :16.0  
 Mean   :6.993   Mean   :15.8  
 3rd Qu.:8.000   3rd Qu.:23.0  
 Max.   :9.000   Max.   :31.0  
                               

attach(airquality)
ggplot(aes(x = Ozone, y = Temp), data = airquality) + geom_point() +
    geom_smooth(method = "lm", se = F) + labs(y = "Temp (F)",
    x = "Ozone (ppb)")

• Model: Temp = 0.2 Ozone + 69.41
  • The intercept, \(\beta_0\) is 69.41
  • The slope, \(\beta_1\) is 0.2
    • For every 1 ppb increase in ozone, the temperature increases by 0.2\(^{\circ}F\)
• The residual standard error is 6.89
  • This is the difference between predicted and actual temperatures
• The \(R^{2}\) value is 0.48
  • Approximately 48% of the variance in temperature is explained by ozone
• p-value is < \(2.2^{-16}\) which indicates statistical significance for this model

summary(lm(Temp ~ Ozone, data = airquality))

## 
## Call:
## lm(formula = Temp ~ Ozone, data = airquality)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -22.147  -4.858   1.828   4.342  12.328 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 69.41072    1.02971   67.41   <2e-16 ***
## Ozone        0.20081    0.01928   10.42   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.819 on 114 degrees of freedom
##   (37 observations deleted due to missingness)
## Multiple R-squared:  0.4877, Adjusted R-squared:  0.4832 
## F-statistic: 108.5 on 1 and 114 DF,  p-value: < 2.2e-16

• Temperature and ozone are positively correlated: as amount of ozone increases, the temperature also increases
• Wind and ozone are negatively correlated: as amount of ozone increases, the wind decreases
• The \(R^{2}\) value for wind is around 0.36
  • This indicates that the linear regression model for temperature vs ozone does a better job predicting than it does for wind vs ozone

model <- lm(airquality$Solar.R ~ airquality$Ozone)
model

## 
## Call:
## lm(formula = airquality$Solar.R ~ airquality$Ozone)
## 
## Coefficients:
##      (Intercept)  airquality$Ozone  
##         144.6306            0.9542

cor.test(airquality$Ozone, airquality$Wind)

## 
##  Pearson's product-moment correlation
## 
## data:  airquality$Ozone and airquality$Wind
## t = -8.0401, df = 114, p-value = 9.272e-13
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.7063918 -0.4708713
## sample estimates:
##        cor 
## -0.6015465

summary(model)

## 
## Call:
## lm(formula = airquality$Solar.R ~ airquality$Ozone)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -153.577  -65.349   -0.555   68.110  176.011 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      144.6306    13.1749   10.98  < 2e-16 ***
## airquality$Ozone   0.9542     0.2459    3.88 0.000179 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 85.83 on 109 degrees of freedom
##   (42 observations deleted due to missingness)
## Multiple R-squared:  0.1213, Adjusted R-squared:  0.1133 
## F-statistic: 15.05 on 1 and 109 DF,  p-value: 0.0001793