Explain why goodnessoffit tests are always righttailed tests

Explain why goodness-of-fit tests are always right-tailed tests. A)We use a normal distribution where only larger values can lead to rejecting the null. B)We use a binomial distribution where only larger values can lead to rejecting the null. C)We use a 2 distribution where only smaller values can lead to rejecting the null. D)We use a 2 distribution where only larger values can lead to rejecting the null.

Solution

D)We use a 2 distribution where only larger values can lead to rejecting the null.

Explain why goodness-of-fit tests are always right-tailed tests.

The chi-squared test is essentially always a one-sided test. Here is a loose way to think about it: the chi-squared test is basically a \'goodness of fit\' test. Sometimes it is explicitly referred to as such, but even when it\'s not, it is still often in essence a goodness of fit. For example, the chi-squared test of independence on a 2 x 2 frequency table is (sort of) a test of goodness of fit of the first row (column) to the distribution specified by the second row (column), and vice versa, simultaneously. Thus, when the realized chi-squared value is way out on the right tail of it\'s distribution, it indicates a poor fit, and if it is far enough, relative to some pre-specified threshold, we might conclude that it is so poor that we don\'t believe the data are from that reference distribution.

If we were to use the chi-squared test as a two-sided test, we would also be worried if the statistic were too far into the left side of the chi-squared distribution. This would mean that we are worried the fit might be too good. This is simply not something we are typically worried about.