This is part of My notes on R programming on my site https://dataz4s.com
The t-test is widely applied statistical method. Here a few examples of how the t-test can be run in R with the t.test() function.
# Creating a vector
x <- c(5,1,2,3,3,7,8,6,3,8)
mean(x)## [1] 4.6The mean is 4.6.
Let’s see how the ‘equal-to’, the ‘less than’ and the ‘greater than’ hypotheses tests can be run with the t.test() function.
# Hypothesis test of the 'equal to'
# H0: mu = 3
t.test(x, alternative="two.sided", mu = 3, conf.level = 0.95)##
## One Sample t-test
##
## data: x
## t = 1.9863, df = 9, p-value = 0.07827
## alternative hypothesis: true mean is not equal to 3
## 95 percent confidence interval:
## 2.77775 6.42225
## sample estimates:
## mean of x
## 4.6At a 0.05 significance level we would fail to reject H0. The test does not provide sufficient evidence that mu different from 3.
# Hypothesis test of the 'greater than'
# H0: mu >= 3
t.test(x, alternative="less", mu = 3, conf.level = 0.95)##
## One Sample t-test
##
## data: x
## t = 1.9863, df = 9, p-value = 0.9609
## alternative hypothesis: true mean is less than 3
## 95 percent confidence interval:
## -Inf 6.076639
## sample estimates:
## mean of x
## 4.6At a 0.05 significance level we would fail to reject H0. The p-value of 0.96 provides very strong proof that mu is greater than 3.
# Hypothesis test of the 'less than'
# H0: mu <= 3
t.test(x, alternative="greater", mu = 3, conf.level = 0.95)##
## One Sample t-test
##
## data: x
## t = 1.9863, df = 9, p-value = 0.03913
## alternative hypothesis: true mean is greater than 3
## 95 percent confidence interval:
## 3.123361 Inf
## sample estimates:
## mean of x
## 4.6At a 0.05 significance level we would reject H0. The test provides evidence that, at a 0.05 significance level, mu is greater than 3.
We will use the LungCap dataset. Read in data:
# Read in data via read_excel
library(readxl)
LungCapData <- read_excel("C:/Users/Usuario/Documents/dataZ4s/R/MarinLectures/LungCapData.xlsx",
col_types = c("numeric", "numeric", "numeric",
"text", "text", "text"))
# attach(LungCapData)
attach(LungCapData)
The LungCap dataset has
# Viewing the first 6 lines of the dataset
head(LungCapData)## # A tibble: 6 x 6
## LungCap Age Height Smoke Gender Caesarean
## <dbl> <dbl> <dbl> <chr> <chr> <chr>
## 1 6.48 6 62.1 no male no
## 2 10.1 18 74.7 yes female no
## 3 9.55 16 69.7 no female yes
## 4 11.1 14 71 no male no
## 5 4.8 5 56.9 no male no
## 6 6.22 11 58.7 no female no# Summarized
summary(LungCapData)## LungCap Age Height Smoke
## Min. : 0.507 Min. : 3.00 Min. :45.30 Length:725
## 1st Qu.: 6.150 1st Qu.: 9.00 1st Qu.:59.90 Class :character
## Median : 8.000 Median :13.00 Median :65.40 Mode :character
## Mean : 7.863 Mean :12.33 Mean :64.84
## 3rd Qu.: 9.800 3rd Qu.:15.00 3rd Qu.:70.30
## Max. :14.675 Max. :19.00 Max. :81.80
## Gender Caesarean
## Length:725 Length:725
## Class :character Class :character
## Mode :character Mode :character
##
##
## # Mean lung capacity of the tested persons
mean(LungCap)## [1] 7.863148# Mean and median age of the tested
mean(Age)## [1] 12.3269median(Age)## [1] 13
# One-tailed hypothesis test for the mean of lung capacity
# H0: mu < 8
# 95% one-tailed confidence interval for mean lung capacity
t.test(LungCap,mu = 8, alternative = "less", conf.level = 0.95)##
## One Sample t-test
##
## data: LungCap
## t = -1.3842, df = 724, p-value = 0.08336
## alternative hypothesis: true mean is less than 8
## 95 percent confidence interval:
## -Inf 8.025974
## sample estimates:
## mean of x
## 7.863148# Can be shortened
t.test(LungCap,mu = 8, alt = "less", conf = 0.95)##
## One Sample t-test
##
## data: LungCap
## t = -1.3842, df = 724, p-value = 0.08336
## alternative hypothesis: true mean is less than 8
## 95 percent confidence interval:
## -Inf 8.025974
## sample estimates:
## mean of x
## 7.863148So, we would fail to reject H0 failing to reject that the mean is less than than 8.
# Two-tailed hypothesis test
# H0: mu = 8
# Two-tailed confidence interval
t.test(LungCap,mu = 8, alt = "two.sided", conf = 0.95)##
## One Sample t-test
##
## data: LungCap
## t = -1.3842, df = 724, p-value = 0.1667
## alternative hypothesis: true mean is not equal to 8
## 95 percent confidence interval:
## 7.669052 8.057243
## sample estimates:
## mean of x
## 7.863148# The 'alt' argument has "two.sided as default
t.test(LungCap, mu = 8, conf = 0.95)##
## One Sample t-test
##
## data: LungCap
## t = -1.3842, df = 724, p-value = 0.1667
## alternative hypothesis: true mean is not equal to 8
## 95 percent confidence interval:
## 7.669052 8.057243
## sample estimates:
## mean of x
## 7.863148
For more advanced analysis and programming it can be useful to extract individual attributes
# Saving the test to an object
TEST <- t.test(LungCap, mu = 8, conf = 0.95)
# Checking
TEST##
## One Sample t-test
##
## data: LungCap
## t = -1.3842, df = 724, p-value = 0.1667
## alternative hypothesis: true mean is not equal to 8
## 95 percent confidence interval:
## 7.669052 8.057243
## sample estimates:
## mean of x
## 7.863148# View attributes of TEST
attributes(TEST)## $names
## [1] "statistic" "parameter" "p.value" "conf.int" "estimate"
## [6] "null.value" "stderr" "alternative" "method" "data.name"
##
## $class
## [1] "htest"# Extracting the test statistic
TEST$statistic## t
## -1.384242# Extracting the p-value
TEST$p.value## [1] 0.1667108
Statslectures by Mick Marin provides a short video on one-sample t-test in R which is the inspiration for the t-test on the Lung Capacity dataset.