tstat: 1-Sample t-Test Statistic
tstatA machine fills milk into 500ml packages. It is suspected that the machine is not working correctly and that the amount of milk filled differs from the setpoint \(\mu_0 = 500\). A sample of \(247\) packages filled by the machine are collected. The sample mean \(\bar{y}\) is equal to \(521.3\) and the sample variance \(s^2_{n-1}\) is equal to \(527.08\).
Test the hypothesis that the amount filled corresponds on average to the setpoint. What is the absolute value of the t-test statistic?
The t-test statistic is calculated by: \[ \begin{aligned} t = \frac{\bar y - \mu_0}{\sqrt{\frac{s^2_{n-1}}{n}}} = \frac{521.3 - 500}{\sqrt{\frac{527.08}{247}}} = 14.581. \end{aligned} \] The absolute value of the t-test statistic is thus equal to 14.581.
A machine fills milk into 500ml packages. It is suspected that the machine is not working correctly and that the amount of milk filled differs from the setpoint \(\mu_0 = 500\). A sample of \(150\) packages filled by the machine are collected. The sample mean \(\bar{y}\) is equal to \(533.7\) and the sample variance \(s^2_{n-1}\) is equal to \(653.07\).
Test the hypothesis that the amount filled corresponds on average to the setpoint. What is the absolute value of the t-test statistic?
The t-test statistic is calculated by: \[ \begin{aligned} t = \frac{\bar y - \mu_0}{\sqrt{\frac{s^2_{n-1}}{n}}} = \frac{533.7 - 500}{\sqrt{\frac{653.07}{150}}} = 16.151. \end{aligned} \] The absolute value of the t-test statistic is thus equal to 16.151.
A machine fills milk into 1000ml packages. It is suspected that the machine is not working correctly and that the amount of milk filled differs from the setpoint \(\mu_0 = 1000\). A sample of \(129\) packages filled by the machine are collected. The sample mean \(\bar{y}\) is equal to \(963.4\) and the sample variance \(s^2_{n-1}\) is equal to \(341.58\).
Test the hypothesis that the amount filled corresponds on average to the setpoint. What is the absolute value of the t-test statistic?
The t-test statistic is calculated by: \[ \begin{aligned} t = \frac{\bar y - \mu_0}{\sqrt{\frac{s^2_{n-1}}{n}}} = \frac{963.4 - 1000}{\sqrt{\frac{341.58}{129}}} = -22.492. \end{aligned} \] The absolute value of the t-test statistic is thus equal to 22.492.
(Note that the HTML output contains mathematical equations in MathML, rendered by MathJax using ‘mathjax = TRUE’. Instead it is also possible to use ‘converter = “pandoc-mathjax”’ so that LaTeX equations are rendered by MathJax directly.)
Demo code:
library("exams")
set.seed(403)
exams2html("tstat.Rmd", mathjax = TRUE)
set.seed(403)
exams2pdf("tstat.Rmd")
set.seed(403)
exams2html("tstat.Rnw", mathjax = TRUE)
set.seed(403)
exams2pdf("tstat.Rnw")