A survey is conducted to test the association between age group (young, middle-aged, elderly) and preference for a political party (A, B, C). What is the degree of freedom for this Chi Square Test?

A researcher wants to use the ${\chi }^2$-test to analyze survey responses divided into four categories. Which of the following conditions must be met for the test to be valid?

Which of the following best describes how the appearance of the ${\chi }^2$ distribution changes as the number of degrees of freedom increases from $2$ to $30$?

A marketing firm conducted a survey of individuals who use smartphones for more than $4$ hours daily. The table below shows the number of such users categorized by age group and type of battery usage.

Perform a ${\chi }^2$ Test of Independence at $\alpha =0.05$ to determine whether primary smartphone battery usage is dependent on age group if ${\chi }^2=0.841$.

Find the critical values ${\chi }_R^2$ and ${\chi }_L^2$ for a confidence level of $c=0.90$ with a sample size of $n=21$.