How to Calculate Critical t-Value in R ? - GeeksforGeeks

08 May, 2021
A critical-T value is a “cut-off point” on the t distribution. A t-distribution is a probability distribution that is used to calculate population parameters when the sample size is small and when the population variance is unknown. T values are used to analyze whether to support or reject a null hypothesis. 
After conducting a t-test, we get its statistics as result. In order to determine the significance of the result, we compare the t- score obtained by a critical t value.  If the absolute value of the t-score is greater than the t critical value, then the results of the test are statistically significant.
Formula:
t = [ x̄ - μ ] / [ s / sqrt( n ) ]
where,
t = t score
x̄ = sample mean,
μ = population mean,
s = standard deviation of the sample,
n = sample size 
Function used:
In order to find the T Critical value we make use of qt() function provided in R Programming Language. 
Syntax: qt(p=conf_value, df= df_value, lower.tail=True/False)
Parameters:
p:- Confidence level
df: degrees of freedom
lower.tail: If TRUE, the probability to the left of p in the t distribution is returned. If FALSE, the probability to the right is returned. By default  it’s value is TRUE.
There are three methods for calculating critical t value, all of them are discussed below:
A right-tailed test is a test in which the hypothesis statement contains a greater than (>) symbol i.e. the inequality points to the right. Sometimes it is also referred to as the upper test.
Here we are assuming a confidence value of 96% i.e.. p= .04 and degree of freedom 4 i.e.. df=4. We are also using the format() function to reduce the decimal value to three decimal places. For the Right tail test, we are setting the value of lower.tail as FALSE.
Example:
R
rm(list = ls())  conf<-.04daf<-4  value<-formatC(qt(p=conf, df=daf, lower.tail=FALSE))      print(paste("Critical T value is : ",value))
Output:
Critical T value is : 2.333
The t critical value is 2.333. Thus, if the test score is greater than this value, the results of the test are statistically significant.
A left-tailed test is a test in which the hypothesis statement contains a less than (<) symbol i.e... the inequality points to the left. Sometimes it is also referred to as the lower test.
Here we are assuming a confidence value of 95% i.e.. p= .05 and degree of freedom 4 i.e.. df=4.  We are also using the format() function to reduce the decimal value to three decimal places. For the Left tail test, we are setting the value of lower.tail as TRUE.
Example:
R
rm(list = ls())  conf<-.05daf<-4  value<-formatC(qt(p=conf, df=daf, lower.tail=TRUE))    print(paste("Critical T value is : ",value))
Output:
Critical T value is : -2.132
The t critical value is -2.132. Thus, if the test score is less than this value, the results of the test are statistically significant.
A two-tailed test is a test in which the hypothesis statement contains both a greater than (>) symbol and a less-than symbol(<) i.e. the inequality points between a certain range. 
In a two-tailed test, we simply need to pass in half of our confidence level in “p” parameter. Here we are assuming confidence value of 96% i.e.. p= .04 and degree of freedom 4 i.e.. df=4.  We are also using the format() function to reduce the decimal value to three decimal places.
Example
R
rm(list = ls())  conf=0.04 / 2  daf<-4  value<-formatC(qt(p = conf , df = daf))       print(paste("Critical T value is : ",value))
Output:
Critical T value is :  -2.999
Whenever, we perform a two-tailed test we get two critical values as output. So here in the above code, the T critical values are 2.999 and -2.999. Therefore, if the test score is less than -2.999 or greater than 2.999, the results of the test are statistically significant.
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