How to Calculate Critical t-Value in R ?

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:

Method 1: Right tailed test

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:

rm(list = ls())

conf<-.04
daf<-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.

Method 2: Left tailed test

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:

rm(list = ls())

conf<-.05
daf<-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.

Method 3: Two-tailed test

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

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.