Theory
Case 1 (denominator \(n\) convention)
Assume that
\[
\begin{cases} U = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \sim \text{N}(0,1) \\ V = \frac{ns^2}{\sigma^2} \sim \chi_{n-1}^2\end{cases}
\]
where
\[
\bar{x} \sim \text{N} \left( \mu, \frac{\sigma^2}{n} \right)
\]
and
\[
\begin{cases} s^2 = \frac{\sum_{i=1}^{n}\left( x_i - \bar{x} \right)^2}{n} \\ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i\end{cases}
\]
and assume that \(U\) and \(V\) are independent.
Since the t-density is defined as
\[
\frac{U}{\sqrt{\frac{V}{n-1}}} = \frac{\text{N}(0,1)}{\sqrt{\frac{\chi_{n-1}^2}{n-1}}} \sim t_{n-1}
\]
it follows that
\[
\begin{align*}\frac{\frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}}{\sqrt{\frac{\frac{ns^2}{\sigma^2}}{n-1}}} &= \frac{\bar{x} -\mu}{\frac{\sigma}{\sqrt{n}}} \times \frac{1}{\frac{s}{\sigma}\sqrt{\frac{n}{n-1}}} \\&= \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n-1}}} \sim t_{n-1}\end{align*}
\]
Case 2 (standard unbiased sample variance)
Case 2 uses the usual unbiased estimator of the variance (denominator \(n-1\) ), which is the convention used in most textbooks and software (including t.test() in R).
Assume that
\[
\begin{cases}U = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \sim \text{N}(0,1) \\V = \frac{(n-1)s^2}{\sigma^2} \sim \chi_{n-1}^2\end{cases}
\]
where
\[
\bar{x} \sim \text{N} \left( \mu, \frac{\sigma^2}{n} \right)
\]
and
\[
\begin{cases}s^2 = \frac{\sum_{i=1}^{n}\left( x_i - \bar{x} \right)^2}{n-1} \\\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i\end{cases}
\]
and assume that \(U\) and \(V\) are independent.
Since the t-density is defined as
\[
\frac{U}{\sqrt{\frac{V}{n-1}}} = \frac{\text{N}(0,1)}{\sqrt{\frac{\chi_{n-1}^2}{n-1}}} \sim t_{n-1}
\]
it follows that
\[
\begin{align*}\frac{\frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}}{\sqrt{\frac{\frac{(n-1)s^2}{\sigma^2}}{n-1}}} &= \frac{\bar{x} -\mu}{\frac{\sigma}{\sqrt{n}}} \times \frac{1}{\frac{s}{\sigma}} \\&= \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \sim t_{n-1}\end{align*}
\]
Software
There are two R modules available to preform the one sample t-test. These are the URLs on the public website:
In RFC these R modules can be found under the “Hypotheses / Empirical Tests” menu item.
To compute the One Sample T-Test on your local machine, the following script can be used in the R console:
set.seed (123 )<- runif (25 , 20 , 40 )# compute confidence intervals = 0.95 #Confidence = 30 #Null Hypothesis <- length (x)<- len - 1 <- sd (x)<- mean (x)<- abs (qt ((1 - par1)/ 2 ,df)) * sd / sqrt (len)<- abs (qt ((1 - par1),df)) * sd / sqrt (len)#Sample size #Sample standard deviation #Sample Mean #2-sided Confidence Interval <- paste ('[' ,mx- delta2)<- paste (dum,',' )<- paste (dum,mx+ delta2)<- paste (dum,']' )#Left-sided Confidence Interval <- paste ('[' ,mx- delta1)<- paste (dum,', +inf ]' )#Right-sided Confidence Interval <- paste ('[ -inf,' ,mx+ delta1)<- paste (dum,']' )# compute two-sided interval and p-value = 'two.sided' = 0.95 #Confidence = 20 #Null Hypothesis <- t.test (x,mu= par3,alternative= par1,conf.level= par2))
[1] 25
[1] 6.010489
[1] 31.91119
[1] "[ 29.4301862828312 , 34.3922024969133 ]"
[1] "[ 29.8545466508001 , +inf ]"
[1] "[ -inf, 33.9678421289444 ]"
One Sample t-test
data: x
t = 9.9087, df = 24, p-value = 5.878e-10
alternative hypothesis: true mean is not equal to 20
95 percent confidence interval:
29.43019 34.39220
sample estimates:
mean of x
31.91119
Practical Example
A sample of intrinsic motivation scores was obtained from students in a statistics course. We wish to test the following hypothesis:
\[
\begin{cases}\text{H}_0: \mu = 19.8 \\\text{H}_A: \mu \neq 19.8\end{cases}
\]
The sample data and computational results are available in the R module shown below. Do we have to accept or reject the Null Hypothesis if we choose a type I error of 3%?
This is a two-sided test because the alternative is pre-specified as \(\text{H}_A:\mu \neq 19.8\) .
Interactive Shiny app (click to load).
Answer: the sample mean \(\bar{x} \simeq 20.06778\) is significantly different from \(\mu_0 = 19.8\) because the p-value is \(0.02471 < 0.03\) .