# tstat2: 1-Sample t-Test Statistic (Single-Choice)

Exercise template for computing the t-test statistic (single-choice) from given hypothesized mean and empirical mean and variance.

**Name:**

`tstat2`

**Type:**

**Related:**

**Preview:**

A machine fills milk into \(500\)ml 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 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 t-test statistic is thus equal to \(14.581\).

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A machine fills milk into \(1000\)ml 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 \(244\) packages filled by the machine are collected. The sample mean \(\bar{y}\) is equal to \(945.9\) and the sample variance \(s^2_{n-1}\) is equal to \(770.7\).

Test the hypothesis that the amount filled corresponds on average to the setpoint. What is the value of the t-test statistic?

The t-test statistic is calculated by: \[ \begin{aligned} t & = & \frac{\bar y - \mu_0}{\sqrt{\frac{s^2_{n-1}}{n}}} = \frac{945.9 - 1000}{\sqrt{\frac{770.7}{244}}} = -30.44. \end{aligned} \] The t-test statistic is thus equal to \(-30.440\).

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A machine fills milk into \(250\)ml packages. It is suspected that the machine is not working correctly and that the amount of milk filled differs from the setpoint \(\mu_0 = 250\). A sample of \(247\) packages filled by the machine are collected. The sample mean \(\bar{y}\) is equal to \(245.1\) and the sample variance \(s^2_{n-1}\) is equal to \(159.93\).

Test the hypothesis that the amount filled corresponds on average to the setpoint. What is the value of the t-test statistic?

The t-test statistic is calculated by: \[ \begin{aligned} t & = & \frac{\bar y - \mu_0}{\sqrt{\frac{s^2_{n-1}}{n}}} = \frac{245.1 - 250}{\sqrt{\frac{159.93}{247}}} = -6.089. \end{aligned} \] The t-test statistic is thus equal to \(-6.089\).

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*(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("tstat2.Rmd", mathjax = TRUE)
set.seed(403)
exams2pdf("tstat2.Rmd")
set.seed(403)
exams2html("tstat2.Rnw", mathjax = TRUE)
set.seed(403)
exams2pdf("tstat2.Rnw")
```