Tests for Independence (College Board AP® Statistics): Study Guide

In an exam, a test for independence may also be referred to as a test for an association between two variables, but be careful with the wording: if two variables are independent, then there is no association between them.

In the exam, either condition is accepted for the large counts condition.

Expected values do not need to be integer values, so leave them unrounded to avoid calculation errors!

Even if you perform the independence test on your calculator, it is still important to show all of your working to demonstrate full understanding. Depending on the question, you may need to show how the chi-square statistic is calculated in full or just how an expected value and the degrees of freedom are calculated.

If you compare the $p$-value with $\alpha$, don't forget that the inequalities are the opposite to when you are comparing the ${\chi }^2$ value to the critical value when you are determining whether or not to reject the null hypothesis!

A coffee company wanted to understand more about who their customers were. They took a random sample of 200 individuals to see if there was an association between an individual's relationship status and whether they were a coffee drinker or not.

The outcomes of their research are shown in the table below.

(a) Determine, at the 5% significance level, if there is an association between an individual's relationship status and whether they are a coffee drinker or not.

(b) After seeing the result of the chi-square test, a researcher concludes that being single causes people to be more likely to drink coffee. Is this a valid conclusion? Justify your answer. Then, using conditional proportions from the data, describe the nature of the association between relationship status and coffee-drinking habits.

Answer:

(a)

Write the null and alternative hypotheses

${\mathrm{H}}_0:$ There is no association between an individual's relationship status and whether they drink coffee or not, they are independent

${\mathrm{H}}_{\mathrm{a}}:$There is an association between an individual's relationship status and whether they drink coffee or not, they are not independent

State the type of test being used

The correct inference procedure is a chi-square test of independence at $\alpha =0.05$

Calculate the expected values (by multiplying the row total by the column total and dividing by the grand total)

Verify the conditions for the test

All conditions for inference have been met:

• The observed values are independent as the sample of individuals is randomly selected

• All expected values are greater than 5

Calculate the chi-square value, ${\chi }^2$, $\sum \frac{{\left(\mathrm{observed}-\mathrm{expected}\right)}^2}{\mathrm{expected}}$

$\begin{array}{rcl}{\chi }^2& =& \frac{{\left(48-39.05\right)}^2}{39.05}+\frac{{\left(25-31.35\right)}^2}{31.35}+\frac{{\left(37-39.6\right)}^2}{39.6}+\frac{{\left(23-31.95\right)}^2}{31.95}+\frac{{\left(32-25.65\right)}^2}{25.65}+\frac{{\left(35-32.4\right)}^2}{32.4}\\ & =& 7.795...\end{array}$

State the number of degrees of freedom

degrees of freedom = $\left(2-1\right)\left(3-1\right)=2$

Find the critical value from the chi-square tables

Find the row corresponding to 2 degrees of freedom and the column corresponding to $\alpha =0.05$

$\mathrm{critical}\mathrm{value}=5.99$

Compare the calculated ${\chi }^2$ value to the critical value and state the conclusion of the test

$\begin{array}{rcl}7.795...& >& 5.99\\ {\chi }^2& >& \mathrm{critical}\mathrm{value}\end{array}$

We reject ${\mathrm{H}}_0$

Interpret the result in the context of the question

There is convincing statistical evidence to suggest that there is an association between an individual's relationship status and whether they are a coffee drinker or not, i.e. the two variables are not independent

(b)

Evaluate the causal claim

The data come from a random sample of 200 individuals, not from a randomized experiment — the researcher did not randomly assign individuals to a relationship status

Because this is an observational study, confounding variables (e.g. age, working hours, lifestyle) could explain the observed association, and a cause-and-effect relationship cannot be established

No, the researcher's causal conclusion is not valid

The chi-square test provides evidence of an association between relationship status and coffee-drinking, but because the study is observational and individuals were not randomly assigned to a relationship status, the data cannot establish that being single causes higher coffee-drinking rates

Compute conditional proportions for coffee drinkers by relationship status

Single: $\frac{48}{71}\approx 0.676$, or 67.6%

Married / Cohabiting: $\frac{25}{57}\approx 0.439$, or 43.9%

Other: $\frac{37}{72}\approx 0.514$, or 51.4%

Describe the nature of the association

The proportion of coffee drinkers is highest among single individuals (about 67.6%), lower among individuals categorized as other (about 51.4%), and lowest among married or cohabiting individuals (about 43.9%)

The sample data suggest that being single is associated with a higher likelihood of being a coffee drinker, while being married or cohabiting is associated with a lower likelihood of being a coffee drinker