Transit Railroads is interested in the relationship between travel distance and the ticket class purchased. A random sample of $200$passengers is taken. Table 11.31 shows the results. The railroad wants to know if a passenger’s choice in ticket class is independent of the distance they must travel.

$\begin{array}{|lllll|}\hline \text{Traveling Distance}& \text{Third class}& \text{Second class}& \text{First class}& \text{Total}\\ 1-100\text{miles}& 21& 14& 6& 41\\ 101-200\text{miles}& 18& 16& 8& 42\\ 201-300\text{miles}& 16& 17& 15& 48\\ 301-400\text{miles}& 12.00\%& 14& 21& 47\\ 401-500\text{miles}& 6& 6& 10& 22\\ \text{Total}& 73& 67& 60& 200\\ \hline\end{array}$

State the hypothesis:

$H_0$:

$H_{\alpha }$:

Given data:

$\begin{array}{|lllll|}\hline \text{Traveling Distance}& \text{Third class}& \text{Second class}& \text{First class}& \text{Total}\\ 1-100\text{miles}& 21& 14& 6& 41\\ 101-200\text{miles}& 18& 16& 8& 42\\ 201-300\text{miles}& 16& 17& 15& 48\\ 301-400\text{miles}& 12.00\%& 14& 21& 47\\ 401-500\text{miles}& 6& 6& 10& 22\\ \text{Total}& 73& 67& 60& 200\\ \hline\end{array}$

The null hypothesis is $H_0$. It's a declaration that there's no difference between sample means or proportions, or between a sample mean or proportion and the population mean or proportion. To put it another way, the difference is zero.

The alternative hypothesis is $H_{\alpha }$: It's a population claim that goes against the null hypothesis and what we conclude when we reject $H_0$.

We conclude that the passenger's choice of ticket class is unaffected by the distance they must travel, as defined by the null hypothesis. We conclude that the passenger's choice of ticket class is influenced by the distance they must go, as defined by the definition of alternate hypothesis.