download.file("http://www.tpmltd.com/ban/nc.RData", destfile = "nc.RData")
load("nc.RData")
summary(nc)
fage mage mature weeks
Min. :14.0 Min. :13 mature mom :133 Min. :20.0
1st Qu.:25.0 1st Qu.:22 younger mom:867 1st Qu.:37.0
Median :30.0 Median :27 Median :39.0
Mean :30.3 Mean :27 Mean :38.3
3rd Qu.:35.0 3rd Qu.:32 3rd Qu.:40.0
Max. :55.0 Max. :50 Max. :45.0
NA's :171 NA's :2
premie visits marital gained
full term:846 Min. : 0.0 married :386 Min. : 0.0
premie :152 1st Qu.:10.0 not married:613 1st Qu.:20.0
NA's : 2 Median :12.0 NA's : 1 Median :30.0
Mean :12.1 Mean :30.3
3rd Qu.:15.0 3rd Qu.:38.0
Max. :30.0 Max. :85.0
NA's :9 NA's :27
weight lowbirthweight gender habit
Min. : 1.00 low :111 female:503 nonsmoker:873
1st Qu.: 6.38 not low:889 male :497 smoker :126
Median : 7.31 NA's : 1
Mean : 7.10
3rd Qu.: 8.06
Max. :11.75
whitemom
not white:284
white :714
NA's : 2
The cases are births recorded in North Carolina. There are 1,000 cases with 13 variables in our sample.
dim(nc)
[1] 1000 13
The medians of the babies for smoker and nonsmoker moms are realitively close, though the median weight for the smoker mom is lower. The spread is relatively similar as well (1.62 vs. 1.657) but still reflects lower birth weight for babies whose moms are smokers. There are more outliers for babies from non smoking moms as well. The plot tells me that babies born to non smoking moms have higher birth weight over all and with much more varibilities in their weights (some outliers for on both sides). Birth weights for both groups are approximately normal.
by(nc$weight, nc$habit, summary)
nc$habit: nonsmoker
Min. 1st Qu. Median Mean 3rd Qu. Max.
1.00 6.44 7.31 7.14 8.06 11.80
--------------------------------------------------------
nc$habit: smoker
Min. 1st Qu. Median Mean 3rd Qu. Max.
1.69 6.08 7.06 6.83 7.74 9.19
boxplot(nc$weight[nc$habit =="nonsmoker"],nc$weight[nc$habit == "smoker"],xlab="habit",ylab="weight",main="Weight vs. habit", names = c("nonsmoker", "smoker"))
by(nc$weight, nc$habit, length)
nc$habit: nonsmoker
[1] 873
--------------------------------------------------------
nc$habit: smoker
[1] 126
I assume the conditions are met: The population consists of birth records for the state of NC. The 1,000 The cases is probably less than 10% of the population to ensure they are simple random samples so independence is reasonable; there are more than 30 mean differences for each category. Even though the distribution of differences shown in the boxplots are a bit skewed they seems reasonable for the size of the sample.
inference(data = nc$weight, group = nc$habit, est = "mean", type = "ci", null = 0,alternative = "twosided", method = "theoretical",order = c("smoker","nonsmoker"))
Warning: package 'openintro' was built under R version 3.1.1
Please visit openintro.org for free statistics materials
Attaching package: 'openintro'
The following object is masked from 'package:datasets':
cars
Warning: package 'BHH2' was built under R version 3.1.1
Attaching package: 'BHH2'
The following object is masked from 'package:openintro':
dotPlot
One quantitative and one categorical variable
Difference between two means
n_smoker = 126 ; n_nonsmoker = 873
Observed difference between means = -0.3155
Standard error = 0.1338
95 % Confidence interval = ( -0.58 , -0.05 )
inference(data = nc$weeks,est = "mean", type = "ci", null = 0,alternative = "twosided", method = "theoretical")
One quantitative variable
Single mean
Observed mean = 38.3347
Standard error = 0.0928
95 % Confidence interval = ( 38.15 , 38.52 )
We are 95 % Confident that the average length of pregnancies (weeks) is between 38.15 and 38.52. This confidence interval is calculated based on the total sample that consists both smoking and nonsmoking mothers.
inference(data = nc$weeks,est = "mean", type = "ci", null = 0,conflevel = 0.90, alternative = "twosided", method = "theoretical")
One quantitative variable
Single mean
Observed mean = 38.3347
Standard error = 0.0928
90 % Confidence interval = ( 38.18 , 38.49 )
The null hypothesis is the average weight gained by younger mothers are not different from the average weight gained by mature mothers. Based on the test statics of p-value of 0.8526 there is strong evidence that we fail to reject the null hypothesis that the average weight gained by younger mothers are not different from the average weight gained by mature mothers.
inference(data = nc$weight, group = nc$mature, est = "mean", type = "ht", null = 0, alternative = "twosided", method = "theoretical")
One quantitative and one categorical variable
Difference between two means
n_mature mom = 133 ; n_younger mom = 867
Observed difference between means = 0.0283
H0: mu_mature mom - mu_younger mom = 0
HA: mu_mature mom - mu_younger mom != 0
Standard error = 0.152
Test statistic: Z = 0.186
p-value: 0.8526
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5.463e-07 5.178e-07 4.907e-07
Based on the summary data and the boxplot the age cutoff for younger and mature mothers is 35. Look at the summary data the max age for younger mom is 34, whereas the mininum age for mature mother is 35. In the boxplot for the age of mature mom the minimum value of a data set is 35.
by(nc$mage, nc$mature,summary)
nc$mature: mature mom
Min. 1st Qu. Median Mean 3rd Qu. Max.
35.0 35.0 37.0 37.2 38.0 50.0
--------------------------------------------------------
nc$mature: younger mom
Min. 1st Qu. Median Mean 3rd Qu. Max.
13.0 21.0 25.0 25.4 30.0 34.0
boxplot(nc$mage[nc$mature == "mature mom"])
We are interested in researching the relationship between mothers' ages and the term of their pregnancy. The null hypothesis will be that there is no relationship between mothers' ages and the term of their pregnancy.
inference(data = nc$mage, group = nc$premie, est = "mean", type = "ht", null = 0,alternative = "twosided", method = "theoretical",order = c("full term","premie"))
One quantitative and one categorical variable
Difference between two means
n_full term = 846 ; n_premie = 152
Observed difference between means = 0.125
H0: mu_full term - mu_premie = 0
HA: mu_full term - mu_premie != 0
Standard error = 0.57
Test statistic: Z = 0.219
p-value: 0.8266
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2.399 2.409 2.42 2.43 2.441 2.451 2.462 2.472 2.483 2.493 2.504 2.515 2.525 2.536 2.546 2.557 2.567 2.578 2.588 2.599 2.609 2.62 2.63 2.641 2.651 2.662 2.672 2.683 2.693 2.704 2.714 2.725 2.735 2.746 2.757 2.767 2.778 2.788 2.799 2.809 2.82 2.83 2.841 2.851 2.862 2.872 2.883 2.893 2.904 2.914 2.925 2.935 2.946 2.956 2.967 2.977 2.988 2.998 3.009 3.02 3.03 3.041 3.051 3.062 3.072 3.083 3.093 3.104 3.114 3.125 3.135 3.146 3.156 3.167 3.177 3.188 3.198 3.209 3.219 3.23 3.24 3.251 3.262 3.272 3.283 3.293 3.304 3.314 3.325 3.335 3.346 3.356 3.367 3.377 3.388 3.398 3.409 3.419 3.43 3.44 3.451 3.461 3.472 3.482 3.493 3.504 3.514 3.525 3.535 3.546 3.556 3.567 3.577 3.588 3.598 3.609 3.619 3.63 3.64 3.651 3.661 3.672 3.682 3.693 3.703 3.714 3.724 3.735 3.745 3.756 3.767 3.777 3.788 3.798 3.809 3.819 3.83 3.84 3.851 3.861 3.872 3.882 3.893 3.903 3.914 3.924 3.935 3.945 3.956 3.966 3.977 3.987 3.998 4.009 4.019 4.03 4.04 4.051 4.061 4.072 4.082 4.093 4.103 4.114 4.124 4.135 4.145 4.156 4.166 4.177 4.187 4.198 4.208 4.219 4.229 4.24 4.251 4.261 4.272 4.282 4.293 4.303 4.314 4.324 4.335 4.345 4.356 4.366 4.377 4.387 4.398 4.408 4.419 4.429 4.44 4.45 4.461 4.471 4.482 4.492 4.503 4.514 4.524 4.535 4.545 4.556 4.566 4.577 4.587 4.598 4.608 4.619 4.629 4.64 4.65 4.661 4.671 4.682 4.692 4.703 4.713 4.724 4.734 4.745 4.756 4.766 4.777 4.787 4.798 4.808 4.819 4.829 4.84 4.85 4.861 4.871 4.882 4.892 4.903 4.913 4.924 4.934 4.945 4.955 4.966 4.976 4.987 4.997 5.008 5.019 5.029 5.04 5.05 5.061 5.071 5.082 5.092 5.103 5.113 5.124 5.134 5.145 5.155 5.166 5.176 5.187 5.197 5.208 5.218 5.229 5.239 5.254.907e-07 5.178e-07 5.463e-07 5.763e-07 6.08e-07 6.413e-07 6.764e-07 7.133e-07 7.521e-07 7.93e-07 8.361e-07 8.813e-07 9.29e-07 9.791e-07 1.032e-06 1.087e-06 1.145e-06 1.207e-06 1.271e-06 1.339e-06 1.41e-06 1.485e-06 1.564e-06 1.646e-06 1.733e-06 1.824e-06 1.92e-06 2.021e-06 2.126e-06 2.237e-06 2.354e-06 2.476e-06 2.605e-06 2.74e-06 2.881e-06 3.03e-06 3.186e-06 3.349e-06 3.521e-06 3.701e-06 3.89e-06 4.088e-06 4.295e-06 4.513e-06 4.741e-06 4.98e-06 5.231e-06 5.494e-06 5.769e-06 6.058e-06 6.361e-06 6.677e-06 7.009e-06 7.357e-06 7.721e-06 8.102e-06 8.501e-06 8.919e-06 9.357e-06 9.815e-06 1.029e-05 1.08e-05 1.132e-05 1.187e-05 1.244e-05 1.305e-05 1.367e-05 1.433e-05 1.502e-05 1.574e-05 1.649e-05 1.727e-05 1.809e-05 1.895e-05 1.985e-05 2.079e-05 2.176e-05 2.279e-05 2.385e-05 2.497e-05 2.614e-05 2.735e-05 2.862e-05 2.995e-05 3.133e-05 3.278e-05 3.428e-05 3.586e-05 3.75e-05 3.921e-05 4.1e-05 4.286e-05 4.48e-05 4.683e-05 4.894e-05 5.114e-05 5.344e-05 5.583e-05 5.833e-05 6.093e-05 6.363e-05 6.646e-05 6.94e-05 7.246e-05 7.565e-05 7.897e-05 8.243e-05 8.603e-05 8.978e-05 9.368e-05 9.774e-05 0.000102 0.0001064 0.0001109 0.0001157 0.0001207 0.0001258 0.0001312 0.0001367 0.0001425 0.0001486 0.0001548 0.0001613 0.0001681 0.0001751 0.0001824 0.00019 0.0001979 0.0002061 0.0002146 0.0002234 0.0002326 0.0002421 0.000252 0.0002623 0.0002729 0.000284 0.0002954 0.0003073 0.0003197 0.0003325 0.0003458 0.0003595 0.0003738 0.0003886 0.000404 0.0004199 0.0004364 0.0004535 0.0004712 0.0004895 0.0005085 0.0005282 0.0005486 0.0005698 0.0005916 0.0006143 0.0006377 0.000662 0.0006871 0.0007131 0.00074 0.0007679 0.0007967 0.0008265 0.0008573 0.0008892 0.0009221 0.0009562 0.0009915 0.001028 0.001066 0.001104 0.001145 0.001186 0.001229 0.001273 0.001319 0.001366 0.001415 0.001466 0.001518 0.001572 0.001627 0.001684 0.001743 0.001804 0.001867 0.001932 0.001999 0.002068 0.002139 0.002212 0.002288 0.002366 0.002446 0.002529 0.002614 0.002702 0.002793 0.002886 0.002982 0.003081 0.003183 0.003288 0.003396 0.003507 0.003622 0.00374 0.003861 0.003985 0.004114 0.004246 0.004381 0.004521 0.004664 0.004812 0.004963 0.005119 0.005279 0.005444 0.005613 0.005787 0.005966 0.006149 0.006337 0.006531 0.006729 0.006933 0.007142 0.007357 0.007577 0.007803 0.008035 0.008273 0.008517 0.008768 0.009025 0.009288 0.009558 0.009835 0.01012 0.01041 0.01071 0.01101 0.01132 0.01164 0.01197 0.01231 0.01265 0.01301 0.01337 0.01373 0.01411 0.0145 0.01489 0.0153 0.01571 0.01614 0.01657 0.01701 0.01747 0.01793 0.0184 0.01888 0.01938 0.01988 0.0204 0.02093 0.02146 0.02201 0.02257 0.02315 0.02373 0.02433 0.02494 0.02556 0.0262 0.02684 0.0275 0.02818 0.02886 0.02956 0.03028 0.03101 0.03175 0.0325 0.03327 0.03406 0.03486 0.03567 0.0365 0.03735 0.03821 0.03908 0.03998 0.04088 0.04181 0.04275 0.0437 0.04467 0.04566 0.04667 0.04769 0.04873 0.04979 0.05086 0.05196 0.05307 0.05419 0.05534 0.0565 0.05768 0.05888 0.0601 0.06134 0.06259 0.06387 0.06516 0.06647 0.0678 0.06915 0.07052 0.07191 0.07331 0.07474 0.07619 0.07765 0.07914 0.08064 0.08217 0.08371 0.08527 0.08686 0.08846 0.09008 0.09172 0.09338 0.09506 0.09677 0.09849 0.1002 0.102 0.1038 0.1056 0.1074 0.1092 0.1111 0.113 0.1148 0.1168 0.1187 0.1206 0.1226 0.1246 0.1266 0.1286 0.1307 0.1327 0.1348 0.1369 0.139 0.1411 0.1433 0.1455 0.1476 0.1498 0.152 0.1543 0.1565 0.1588 0.161 0.1633 0.1656 0.1679 0.1703 0.1726 0.175 0.1773 0.1797 0.1821 0.1845 0.1869 0.1894 0.1918 0.1942 0.1967 0.1992 0.2016 0.2041 0.2066 0.2091 0.2116 0.2141 0.2166 0.2191 0.2217 0.2242 0.2267 0.2293 0.2318 0.2343 0.2369 0.2394 0.242 0.2445 0.2471 0.2496 0.2521 0.2547 0.2572 0.2598 0.2623 0.2648 0.2673 0.2698 0.2723 0.2748 0.2773 0.2798 0.2823 0.2848 0.2872 0.2896 0.2921 0.2945 0.2969 0.2993 0.3017 0.304 0.3064 0.3087 0.311 0.3133 0.3156 0.3179 0.3201 0.3224 0.3246 0.3267 0.3289 0.331 0.3332 0.3352 0.3373 0.3394 0.3414 0.3434 0.3453 0.3473 0.3492 0.3511 0.3529 0.3547 0.3565 0.3583 0.36 0.3617 0.3634 0.365 0.3666 0.3682 0.3697 0.3712 0.3727 0.3741 0.3755 0.3768 0.3782 0.3794 0.3807 0.3819 0.3831 0.3842 0.3853 0.3863 0.3873 0.3883 0.3892 0.3901 0.3909 0.3917 0.3925 0.3932 0.3939 0.3945 0.3951 0.3957 0.3962 0.3966 0.3971 0.3974 0.3978 0.398 0.3983 0.3985 0.3986 0.3988 0.3988 0.3988 0.3988 0.3988 0.3986 0.3985 0.3983 0.398 0.3978 0.3974 0.3971 0.3966 0.3962 0.3957 0.3951 0.3945 0.3939 0.3932 0.3925 0.3917 0.3909 0.3901 0.3892 0.3883 0.3873 0.3863 0.3853 0.3842 0.3831 0.3819 0.3807 0.3794 0.3782 0.3768 0.3755 0.3741 0.3727 0.3712 0.3697 0.3682 0.3666 0.365 0.3634 0.3617 0.36 0.3583 0.3565 0.3547 0.3529 0.3511 0.3492 0.3473 0.3453 0.3434 0.3414 0.3394 0.3373 0.3352 0.3332 0.331 0.3289 0.3267 0.3246 0.3224 0.3201 0.3179 0.3156 0.3133 0.311 0.3087 0.3064 0.304 0.3017 0.2993 0.2969 0.2945 0.2921 0.2896 0.2872 0.2848 0.2823 0.2798 0.2773 0.2748 0.2723 0.2698 0.2673 0.2648 0.2623 0.2598 0.2572 0.2547 0.2521 0.2496 0.2471 0.2445 0.242 0.2394 0.2369 0.2343 0.2318 0.2293 0.2267 0.2242 0.2217 0.2191 0.2166 0.2141 0.2116 0.2091 0.2066 0.2041 0.2016 0.1992 0.1967 0.1942 0.1918 0.1894 0.1869 0.1845 0.1821 0.1797 0.1773 0.175 0.1726 0.1703 0.1679 0.1656 0.1633 0.161 0.1588 0.1565 0.1543 0.152 0.1498 0.1476 0.1455 0.1433 0.1411 0.139 0.1369 0.1348 0.1327 0.1307 0.1286 0.1266 0.1246 0.1226 0.1206 0.1187 0.1168 0.1148 0.113 0.1111 0.1092 0.1074 0.1056 0.1038 0.102 0.1002 0.09849 0.09677 0.09506 0.09338 0.09172 0.09008 0.08846 0.08686 0.08527 0.08371 0.08217 0.08064 0.07914 0.07765 0.07619 0.07474 0.07331 0.07191 0.07052 0.06915 0.0678 0.06647 0.06516 0.06387 0.06259 0.06134 0.0601 0.05888 0.05768 0.0565 0.05534 0.05419 0.05307 0.05196 0.05086 0.04979 0.04873 0.04769 0.04667 0.04566 0.04467 0.0437 0.04275 0.04181 0.04088 0.03998 0.03908 0.03821 0.03735 0.0365 0.03567 0.03486 0.03406 0.03327 0.0325 0.03175 0.03101 0.03028 0.02956 0.02886 0.02818 0.0275 0.02684 0.0262 0.02556 0.02494 0.02433 0.02373 0.02315 0.02257 0.02201 0.02146 0.02093 0.0204 0.01988 0.01938 0.01888 0.0184 0.01793 0.01747 0.01701 0.01657 0.01614 0.01571 0.0153 0.01489 0.0145 0.01411 0.01373 0.01337 0.01301 0.01265 0.01231 0.01197 0.01164 0.01132 0.01101 0.01071 0.01041 0.01012 0.009835 0.009558 0.009288 0.009025 0.008768 0.008517 0.008273 0.008035 0.007803 0.007577 0.007357 0.007142 0.006933 0.006729 0.006531 0.006337 0.006149 0.005966 0.005787 0.005613 0.005444 0.005279 0.005119 0.004963 0.004812 0.004664 0.004521 0.004381 0.004246 0.004114 0.003985 0.003861 0.00374 0.003622 0.003507 0.003396 0.003288 0.003183 0.003081 0.002982 0.002886 0.002793 0.002702 0.002614 0.002529 0.002446 0.002366 0.002288 0.002212 0.002139 0.002068 0.001999 0.001932 0.001867 0.001804 0.001743 0.001684 0.001627 0.001572 0.001518 0.001466 0.001415 0.001366 0.001319 0.001273 0.001229 0.001186 0.001145 0.001104 0.001066 0.001028 0.0009915 0.0009562 0.0009221 0.0008892 0.0008573 0.0008265 0.0007967 0.0007679 0.00074 0.0007131 0.0006871 0.000662 0.0006377 0.0006143 0.0005916 0.0005698 0.0005486 0.0005282 0.0005085 0.0004895 0.0004712 0.0004535 0.0004364 0.0004199 0.000404 0.0003886 0.0003738 0.0003595 0.0003458 0.0003325 0.0003197 0.0003073 0.0002954 0.000284 0.0002729 0.0002623 0.000252 0.0002421 0.0002326 0.0002234 0.0002146 0.0002061 0.0001979 0.00019 0.0001824 0.0001751 0.0001681 0.0001613 0.0001548 0.0001486 0.0001425 0.0001367 0.0001312 0.0001258 0.0001207 0.0001157 0.0001109 0.0001064 0.000102 9.774e-05 9.368e-05 8.978e-05 8.603e-05 8.243e-05 7.897e-05 7.565e-05 7.246e-05 6.94e-05 6.646e-05 6.363e-05 6.093e-05 5.833e-05 5.583e-05 5.344e-05 5.114e-05 4.894e-05 4.683e-05 4.48e-05 4.286e-05 4.1e-05 3.921e-05 3.75e-05 3.586e-05 3.428e-05 3.278e-05 3.133e-05 2.995e-05 2.862e-05 2.735e-05 2.614e-05 2.497e-05 2.385e-05 2.279e-05 2.176e-05 2.079e-05 1.985e-05 1.895e-05 1.809e-05 1.727e-05 1.649e-05 1.574e-05 1.502e-05 1.433e-05 1.367e-05 1.305e-05 1.244e-05 1.187e-05 1.132e-05 1.08e-05 1.029e-05 9.815e-06 9.357e-06 8.919e-06 8.501e-06 8.102e-06 7.721e-06 7.357e-06 7.009e-06 6.677e-06 6.361e-06 6.058e-06 5.769e-06 5.494e-06 5.231e-06 4.98e-06 4.741e-06 4.513e-06 4.295e-06 4.088e-06 3.89e-06 3.701e-06 3.521e-06 3.349e-06 3.186e-06 3.03e-06 2.881e-06 2.74e-06 2.605e-06 2.476e-06 2.354e-06 2.237e-06 2.126e-06 2.021e-06 1.92e-06 1.824e-06 1.733e-06 1.646e-06 1.564e-06 1.485e-06 1.41e-06 1.339e-06 1.271e-06 1.207e-06 1.145e-06 1.087e-06 1.032e-06 9.791e-07 9.29e-07 8.813e-07 8.361e-07 7.93e-07 7.521e-07 7.133e-07 6.764e-07 6.413e-07 6.08e-07 5.763e-07 5.463e-07 5.178e-07 4.907e-07
based on the test statics of p-value of 0.8266 there is strong evidence that we fail to reject the null hypothesis there is no relationship between mothers' ages and the term of their pregnancy.
meandiff <- 1.1
sddiff <- 4.9
n<- 51
semeandiff<-sddiff/sqrt(51)
semeandiff
[1] 0.6861
lower <-meandiff - qt(0.95, df=50)*semeandiff
upper <- meandiff + qt(0.95, df=50)*semeandiff
c(lower, upper)
[1] -0.0499 2.2499
Between January 1 of 1968 and 2008, the 90% confidence interval of the average difference in temperature is between -0.0321 and 2.2321.
No, it doesn't. The null hypothesis for this question would be that the difference between the average of two temperature reading is zero. Since the calculated lower confidence level is negative (below zero) and the upper level is positive we fail to reject the null hypothesis as the mean of the temperature differences could be zero.
The distribution of income for Cleveland and Sacramento need to be normal for the t test to be valid. The sample size are small, 21 for cleveland and 17 for Sacramento so we would expect reasonable skewness of the data. First he needs to consider if the two sample means are independent (from less than 10% of the population). Another condition for the distribution to be considered near normal is that the distribution of sample observations is not strongly skewed. Looking at the histograms and the statistical summaries I do not think he should carry forward with the t test because both samples have strong positively skewed outliers.
n x¯ s min max 25 7.73 0.77 6.17 9.78
The observations are simple random sample and consist of less thand 10% of New Yorkers, therefore independence is reasonable. The summary statistics do notsuggest any strong skew or outliers. The normality assumption seems reasonable.
We use the row with 24 degrees of freedom (25 - 1). The value T = 1.753 (ignore the negative sign)
null <- 8
sd <- 0.77
n <- 25
mean <-7.73
t <-(mean-null)/(sd/sqrt(n))
t
[1] -1.753
Based on the degrees of freedom of 24 T value of 1.753 falls between the 2nd of 3rd columns. Since this is a single tail test this corresponds to a p-value between 0.025 and 0.05. The p-value is not guaranteed to be less than 0.05 so we fail to reject the null hypothesis.
These data does not provide strong evidence that New Yorkers sleep less than 8 hours a night on average.
Yes the interval contains the 8 hours.
mean
[1] 7.73
se<-0.77/sqrt(25)
se
[1] 0.154
t24<-2.49
lower<-mean-t24*se
upper <-mean+t24*se
c(lower,upper)
[1] 7.347 8.113
Since the calculated p value is so small we reject the null hypothesis.
Looking at the boxplot the variance for each group is not equal from one to the next. The group for horsebean and sunflower have a much smaller variance than the rest of the group.
observedF <-15.36
pf(observedF, df1 = 5, df2 = 65, lower.tail=FALSE)
[1] 5.968e-10
Three conditions must be checked before ANOVA analsysi:
Since the P value of 0.0003 is much smaller than the assumed significance level of 0.05 we reject the null hypoesis and conclude that the average physical activity level varies among at least some of the different levels of coffee consumption.