Discussion 5
Carola Rojas
2/16/2018
The R datasets library contains data on air quality in New York (airquality). Conduct a hypothesis test to evaluate if ozone levels are a function of month. NOTE: dichotomize month. If that test were significant, what else would be required? Post your hypothesis test and R code here.
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
## In order to find a possible correlation between ozone levels and month one must develop a hypothesis test with a p-value of 0.05, anything smaller than that will mean that you will be able to reject the null hypothesis.
We will first define each month in order to obtain precise findings.
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$Month=="9")
boxplot(airquality$Ozone~airquality$Month)
We are visualizing the levels of ozone per month, I would also want to see if the level of ozone is dependent on temperature which I am assuming it is. The reason why I am looking at temperature is because I am inferring it is the variable that would have the most direct impact on ozone levels per month.
plot(airquality$Ozone~airquality$Temp)
It appears that the higher the temperature is the higher the levels of Ozone. Thus, they appear to have a positive correlation.
We can now see the specific relationship for each month
lmm5<- lm(month5$Ozone~month5$Temp)
summary(lmm5)##
## Call:
## lm(formula = month5$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: 0.003315lmm6<- lm(month6$Ozone~month6$Temp)
summary(lmm6)##
## 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$Temp 1.5524 0.6531 2.377 0.0491 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 14.48 on 7 degrees of freedom
## (21 observations deleted due to missingness)
## Multiple R-squared: 0.4467, Adjusted R-squared: 0.3676
## F-statistic: 5.651 on 1 and 7 DF, p-value: 0.04909lmm7<- lm(month7$Ozone~month7$Temp)
summary(lmm7)##
## Call:
## lm(formula = month7$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-05lmm8<- lm(month8$Ozone~month8$Temp)
summary(lmm8)##
## 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$Temp 3.559 0.974 3.654 0.00126 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 32.46 on 24 degrees of freedom
## (5 observations deleted due to missingness)
## Multiple R-squared: 0.3575, Adjusted R-squared: 0.3307
## F-statistic: 13.35 on 1 and 24 DF, p-value: 0.001256lmm9<- lm(month9$Ozone~month9$Temp)
summary(lmm9)##
## Call:
## lm(formula = month9$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-08While looking at all of this data one can note that all the p-values are significantly smaller than the previously established significance level of 0.05. Which means that with a 95% confidence level one can reject the null hypothesis and conclude that ozone is indeed a function of month and temperature. It is important to note that with the exception of month 6, if we were to establish a larger confidence level, such as 99% and 98%, we would still arrive to the same conclusion.