Three Apps or dashboards created with R package Shiny. It takes a few moments to activate the Apps in the server as they are stopped after 10 min without user interaction.
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1 t-distribution compared to the Normal distribution
As the number of degrees of freedom increases the t distribution approaches the Normal distribution.
The observed t-value is calculated as \({\bar x} / s_{\bar x}\), or more generaly the estimate of the value of a parameter divided by its standard deviation.
2 F-distribution
The observed F-value is the ratio between two estimates of variance \(s^2_1 / s^2_2\). If the variance as numerator is larger than the denominator \(F > 1\). The degrees of freedom of the two variances determine the shape of the distribution.
If we know the observed F-value, and degrees of freedom we can look up the P-value from the theoretical distribution.