library(tidyverse)## -- Attaching packages --------------------------------------- tidyverse 1.3.1 --## v ggplot2 3.3.5 v purrr 0.3.4
## v tibble 3.1.6 v dplyr 1.0.8
## v tidyr 1.2.0 v stringr 1.4.0
## v readr 2.1.2 v forcats 0.5.1## -- Conflicts ------------------------------------------ tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag() masks stats::lag()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 30mydata = data.frame(airquality$Ozone,airquality$Month)
# I first seperated the Ozone and Month Variables from the rest of the data
mydata = na.omit(mydata)
# Then I ommitted all the NA values from this data set
mydata## airquality.Ozone airquality.Month
## 1 41 5
## 2 36 5
## 3 12 5
## 4 18 5
## 6 28 5
## 7 23 5
## 8 19 5
## 9 8 5
## 11 7 5
## 12 16 5
## 13 11 5
## 14 14 5
## 15 18 5
## 16 14 5
## 17 34 5
## 18 6 5
## 19 30 5
## 20 11 5
## 21 1 5
## 22 11 5
## 23 4 5
## 24 32 5
## 28 23 5
## 29 45 5
## 30 115 5
## 31 37 5
## 38 29 6
## 40 71 6
## 41 39 6
## 44 23 6
## 47 21 6
## 48 37 6
## 49 20 6
## 50 12 6
## 51 13 6
## 62 135 7
## 63 49 7
## 64 32 7
## 66 64 7
## 67 40 7
## 68 77 7
## 69 97 7
## 70 97 7
## 71 85 7
## 73 10 7
## 74 27 7
## 76 7 7
## 77 48 7
## 78 35 7
## 79 61 7
## 80 79 7
## 81 63 7
## 82 16 7
## 85 80 7
## 86 108 7
## 87 20 7
## 88 52 7
## 89 82 7
## 90 50 7
## 91 64 7
## 92 59 7
## 93 39 8
## 94 9 8
## 95 16 8
## 96 78 8
## 97 35 8
## 98 66 8
## 99 122 8
## 100 89 8
## 101 110 8
## 104 44 8
## 105 28 8
## 106 65 8
## 108 22 8
## 109 59 8
## 110 23 8
## 111 31 8
## 112 44 8
## 113 21 8
## 114 9 8
## 116 45 8
## 117 168 8
## 118 73 8
## 120 76 8
## 121 118 8
## 122 84 8
## 123 85 8
## 124 96 9
## 125 78 9
## 126 73 9
## 127 91 9
## 128 47 9
## 129 32 9
## 130 20 9
## 131 23 9
## 132 21 9
## 133 24 9
## 134 44 9
## 135 21 9
## 136 28 9
## 137 9 9
## 138 13 9
## 139 46 9
## 140 18 9
## 141 13 9
## 142 24 9
## 143 16 9
## 144 13 9
## 145 23 9
## 146 36 9
## 147 7 9
## 148 14 9
## 149 30 9
## 151 14 9
## 152 18 9
## 153 20 9Ho: The ozone levels are the same each month Ha: The ozone levels are different each month
mydata5 = mydata %>% filter(airquality.Month==5)
N5=26
M5=mean(mydata5$airquality.Ozone)
M5## [1] 23.61538S5=sd(mydata5$airquality.Ozone)
S5## [1] 22.22445mydata6 = mydata %>% filter(airquality.Month==6)
N6=9
M6=mean(mydata6$airquality.Ozone)
M6## [1] 29.44444S6=sd(mydata6$airquality.Ozone)
S6## [1] 18.2079mydata7 = mydata %>% filter(airquality.Month==7)
N7=26
M7=mean(mydata7$airquality.Ozone)
M7## [1] 59.11538S7=sd(mydata7$airquality.Ozone)
S7## [1] 31.63584mydata8 = mydata %>% filter(airquality.Month==8)
N8=26
M8=mean(mydata8$airquality.Ozone)
M8## [1] 59.96154S8=sd(mydata8$airquality.Ozone)
S8## [1] 39.68121mydata9 = mydata %>% filter(airquality.Month==9)
N9=29
M9=mean(mydata9$airquality.Ozone)
M9## [1] 31.44828S9=sd(mydata9$airquality.Ozone)
S9## [1] 24.14182compare the different months to each other and determines if we reject or fail to reject the null hypothesis
((M5-M6)-0)/(sqrt((S5^2/N5)+(S6^2/N6)))## [1] -0.7801009((M5-M7)-0)/(sqrt((S5^2/N5)+(S7^2/N7)))## [1] -4.681989((M5-M8)-0)/(sqrt((S5^2/N5)+(S8^2/N8)))## [1] -4.07488((M5-M9)-0)/(sqrt((S5^2/N5)+(S9^2/N9)))## [1] -1.252747((M6-M7)-0)/(sqrt((S6^2/N6)+(S7^2/N7)))## [1] -3.418598((M6-M8)-0)/(sqrt((S6^2/N6)+(S8^2/N8)))## [1] -3.092206((M6-M9)-0)/(sqrt((S6^2/N6)+(S9^2/N9)))## [1] -0.2655679((M7-M8)-0)/(sqrt((S7^2/N7)+(S8^2/N8)))## [1] -0.08501814((M7-M9)-0)/(sqrt((S7^2/N7)+(S9^2/N9)))## [1] 3.614506((M8-M9)-0)/(sqrt((S8^2/N8)+(S9^2/N9)))## [1] 3.174831There is a difference in ozone between, months: 5-7, 6-8, 5-8, 6-7, 7-9, 8-9.a