week5
Yifeng Qi
2021/3/1
Ho: There is no change in ozones level as the months change. Ha: There is a change in ozones level as the months change.
library(datasets)
datasets::airquality## 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
## 7 23 299 8.6 65 5 7
## 8 19 99 13.8 59 5 8
## 9 8 19 20.1 61 5 9
## 10 NA 194 8.6 69 5 10
## 11 7 NA 6.9 74 5 11
## 12 16 256 9.7 69 5 12
## 13 11 290 9.2 66 5 13
## 14 14 274 10.9 68 5 14
## 15 18 65 13.2 58 5 15
## 16 14 334 11.5 64 5 16
## 17 34 307 12.0 66 5 17
## 18 6 78 18.4 57 5 18
## 19 30 322 11.5 68 5 19
## 20 11 44 9.7 62 5 20
## 21 1 8 9.7 59 5 21
## 22 11 320 16.6 73 5 22
## 23 4 25 9.7 61 5 23
## 24 32 92 12.0 61 5 24
## 25 NA 66 16.6 57 5 25
## 26 NA 266 14.9 58 5 26
## 27 NA NA 8.0 57 5 27
## 28 23 13 12.0 67 5 28
## 29 45 252 14.9 81 5 29
## 30 115 223 5.7 79 5 30
## 31 37 279 7.4 76 5 31
## 32 NA 286 8.6 78 6 1
## 33 NA 287 9.7 74 6 2
## 34 NA 242 16.1 67 6 3
## 35 NA 186 9.2 84 6 4
## 36 NA 220 8.6 85 6 5
## 37 NA 264 14.3 79 6 6
## 38 29 127 9.7 82 6 7
## 39 NA 273 6.9 87 6 8
## 40 71 291 13.8 90 6 9
## 41 39 323 11.5 87 6 10
## 42 NA 259 10.9 93 6 11
## 43 NA 250 9.2 92 6 12
## 44 23 148 8.0 82 6 13
## 45 NA 332 13.8 80 6 14
## 46 NA 322 11.5 79 6 15
## 47 21 191 14.9 77 6 16
## 48 37 284 20.7 72 6 17
## 49 20 37 9.2 65 6 18
## 50 12 120 11.5 73 6 19
## 51 13 137 10.3 76 6 20
## 52 NA 150 6.3 77 6 21
## 53 NA 59 1.7 76 6 22
## 54 NA 91 4.6 76 6 23
## 55 NA 250 6.3 76 6 24
## 56 NA 135 8.0 75 6 25
## 57 NA 127 8.0 78 6 26
## 58 NA 47 10.3 73 6 27
## 59 NA 98 11.5 80 6 28
## 60 NA 31 14.9 77 6 29
## 61 NA 138 8.0 83 6 30
## 62 135 269 4.1 84 7 1
## 63 49 248 9.2 85 7 2
## 64 32 236 9.2 81 7 3
## 65 NA 101 10.9 84 7 4
## 66 64 175 4.6 83 7 5
## 67 40 314 10.9 83 7 6
## 68 77 276 5.1 88 7 7
## 69 97 267 6.3 92 7 8
## 70 97 272 5.7 92 7 9
## 71 85 175 7.4 89 7 10
## 72 NA 139 8.6 82 7 11
## 73 10 264 14.3 73 7 12
## 74 27 175 14.9 81 7 13
## 75 NA 291 14.9 91 7 14
## 76 7 48 14.3 80 7 15
## 77 48 260 6.9 81 7 16
## 78 35 274 10.3 82 7 17
## 79 61 285 6.3 84 7 18
## 80 79 187 5.1 87 7 19
## 81 63 220 11.5 85 7 20
## 82 16 7 6.9 74 7 21
## 83 NA 258 9.7 81 7 22
## 84 NA 295 11.5 82 7 23
## 85 80 294 8.6 86 7 24
## 86 108 223 8.0 85 7 25
## 87 20 81 8.6 82 7 26
## 88 52 82 12.0 86 7 27
## 89 82 213 7.4 88 7 28
## 90 50 275 7.4 86 7 29
## 91 64 253 7.4 83 7 30
## 92 59 254 9.2 81 7 31
## 93 39 83 6.9 81 8 1
## 94 9 24 13.8 81 8 2
## 95 16 77 7.4 82 8 3
## 96 78 NA 6.9 86 8 4
## 97 35 NA 7.4 85 8 5
## 98 66 NA 4.6 87 8 6
## 99 122 255 4.0 89 8 7
## 100 89 229 10.3 90 8 8
## 101 110 207 8.0 90 8 9
## 102 NA 222 8.6 92 8 10
## 103 NA 137 11.5 86 8 11
## 104 44 192 11.5 86 8 12
## 105 28 273 11.5 82 8 13
## 106 65 157 9.7 80 8 14
## 107 NA 64 11.5 79 8 15
## 108 22 71 10.3 77 8 16
## 109 59 51 6.3 79 8 17
## 110 23 115 7.4 76 8 18
## 111 31 244 10.9 78 8 19
## 112 44 190 10.3 78 8 20
## 113 21 259 15.5 77 8 21
## 114 9 36 14.3 72 8 22
## 115 NA 255 12.6 75 8 23
## 116 45 212 9.7 79 8 24
## 117 168 238 3.4 81 8 25
## 118 73 215 8.0 86 8 26
## 119 NA 153 5.7 88 8 27
## 120 76 203 9.7 97 8 28
## 121 118 225 2.3 94 8 29
## 122 84 237 6.3 96 8 30
## 123 85 188 6.3 94 8 31
## 124 96 167 6.9 91 9 1
## 125 78 197 5.1 92 9 2
## 126 73 183 2.8 93 9 3
## 127 91 189 4.6 93 9 4
## 128 47 95 7.4 87 9 5
## 129 32 92 15.5 84 9 6
## 130 20 252 10.9 80 9 7
## 131 23 220 10.3 78 9 8
## 132 21 230 10.9 75 9 9
## 133 24 259 9.7 73 9 10
## 134 44 236 14.9 81 9 11
## 135 21 259 15.5 76 9 12
## 136 28 238 6.3 77 9 13
## 137 9 24 10.9 71 9 14
## 138 13 112 11.5 71 9 15
## 139 46 237 6.9 78 9 16
## 140 18 224 13.8 67 9 17
## 141 13 27 10.3 76 9 18
## 142 24 238 10.3 68 9 19
## 143 16 201 8.0 82 9 20
## 144 13 238 12.6 64 9 21
## 145 23 14 9.2 71 9 22
## 146 36 139 10.3 81 9 23
## 147 7 49 10.3 69 9 24
## 148 14 20 16.6 63 9 25
## 149 30 193 6.9 70 9 26
## 150 NA 145 13.2 77 9 27
## 151 14 191 14.3 75 9 28
## 152 18 131 8.0 76 9 29
## 153 20 223 11.5 68 9 30summary (airquality)## 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
## Dichotomize month
month5<-subset(airquality,airquality$Month=="5")
month6<-subset(airquality,airquality$Month=="6")
month7<-subset(airquality,airquality$Month=="7")
month8<-subset(airquality,airquality$Month=="8")
month9<-subset(airquality,airquality$Month=="9")
plot(airquality$Ozone~airquality$Temp)
Find P-value
linearModelM5<- lm(month5$Ozone~month5$Temp)
summary(linearModelM5)##
## Call:
## lm(formula = month5$Ozone ~ month5$Temp)
##
## Residuals:
## Min 1Q Median 3Q Max
## -30.316 -8.622 -2.411 5.320 68.259
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -102.159 38.750 -2.636 0.01446 *
## month5$Temp 1.885 0.578 3.261 0.00331 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 18.88 on 24 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.307, Adjusted R-squared: 0.2781
## F-statistic: 10.63 on 1 and 24 DF, p-value: 0.003315linearModelM6<- lm(month6$Ozone~month6$Temp)
summary(linearModelM6)##
## Call:
## lm(formula = month6$Ozone ~ month6$Temp)
##
## Residuals:
## Min 1Q Median 3Q Max
## -12.995 -9.337 -6.309 11.082 23.271
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -91.9910 51.3120 -1.793 0.1161
## month6$Temp 1.5524 0.6531 2.377 0.0491 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 14.48 on 7 degrees of freedom
## (21 observations deleted due to missingness)
## Multiple R-squared: 0.4467, Adjusted R-squared: 0.3676
## F-statistic: 5.651 on 1 and 7 DF, p-value: 0.04909linearModelM7<- lm(month7$Ozone~month7$Temp)
summary(linearModelM7)##
## Call:
## lm(formula = month7$Ozone ~ month7$Temp)
##
## Residuals:
## Min 1Q Median 3Q Max
## -32.108 -14.522 -1.161 7.582 75.290
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -372.921 84.453 -4.416 0.000184 ***
## month7$Temp 5.150 1.005 5.123 3.05e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 22.32 on 24 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.5223, Adjusted R-squared: 0.5024
## F-statistic: 26.24 on 1 and 24 DF, p-value: 3.048e-05linearModelM8<- lm(month8$Ozone~month8$Temp)
summary(linearModelM8)##
## Call:
## lm(formula = month8$Ozone ~ month8$Temp)
##
## Residuals:
## Min 1Q Median 3Q Max
## -40.421 -17.651 -8.067 9.974 118.579
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -238.861 82.023 -2.912 0.00764 **
## month8$Temp 3.559 0.974 3.654 0.00126 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 32.46 on 24 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.3575, Adjusted R-squared: 0.3307
## F-statistic: 13.35 on 1 and 24 DF, p-value: 0.001256linearModelM9<- lm(month9$Ozone~month9$Temp)
summary(linearModelM9)##
## Call:
## lm(formula = month9$Ozone ~ month9$Temp)
##
## Residuals:
## Min 1Q Median 3Q Max
## -27.447 -8.585 -3.692 11.041 31.392
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -149.3469 23.6876 -6.305 9.51e-07 ***
## month9$Temp 2.3511 0.3062 7.677 2.95e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.78 on 27 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.6858, Adjusted R-squared: 0.6742
## F-statistic: 58.94 on 1 and 27 DF, p-value: 2.945e-08p-value for Month 5 = 0.003315 p-value for Month 6 = 0.04909 p-value for Month 7 = 3.048e-05 p-value for Month 8 = 0.001256 p-value for Month 9 = 2.945e-08
As we calculated, each monthโs p-value is less than 0.05, so we reject the null hypothesis.