Hypothesis testing with chi-squared test

Question

We asked 460 persons whether they like coffee (yes/no) or tea (yes/no). 416 like both, 5 like none of them, 16 like coffee but do not like tea, and 23 like tea but do not like coffee. How can we decide whether the attitudes toward coffee and tea are independent with level of significance α = 0.05?

Answer 1

Outline toward a solution:

Start by making a $2 \times 2$ table with column headers Tea Yes and Tea No and row headers Coffee Yes and Coffee No. Put the appropriate frequencies 416, 5, etc. into the four cells of the table. Get row totals, column totals, and the grand total (460).

Then check your textbook for the method of computing the chi-squared goodness-of-fit (GOF) statistic. (For this simple two-by-two case, some texts have a different form of the GOF statistic than others, so I won't try to quote the formula here.)

The appropriate degrees of freedom will be df=1. Look up the critical value (3.8415) of a test at the 5% level. [It is the value that cuts 5% from the upper tail of the distribution $Chisq(df=1).$] If your GOF statistic exceeds the critical value, reject the null hypothesis of independence (otherwise, don't reject).