Question

Explain why a chi-square goodness-of-fit test, a chi-square independence test, or a chi-square homogeneity test is always right tailed.

Short answer

For all three tests, the null hypothesis is rejected only when the observed and expected frequencies match up poorly, which corresponds to large values of the chi-square test statistics, resulting in all the three tests are always right tailed.

Step-by-step solution

Step 1. Explain main properties of the chi-square distributions.

The properties of the chi-square distributions is given below,

(i) The research hypothesis for the chi-square is always a one-tailed test. Hence, the distributions are positively skewed.

(ii) Chi-square values are always positive. The minimum possible value is zero, and there is no upper limit to its maximum value. A chi-square of zero means that the variables are completely independent and the observed frequencies in every cell are equal to the corresponding expected frequencies.

(iii) As the number of degrees of freedom increases, the chi-square distribution becomes more symmetrical and with degrees of freedom greater than $30$, begins to resemble the normal curve.

Step 2. Explain the reason.

Considering all the above given properties,

When the observed and expected frequencies match up poorly then only the null hypothesis is rejected, corresponding to larger values of the chi-square test statistic.

Therefore, all the three tests are always right tailed.