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(Example from Khan Academy)

Let's say that we're interested in buying a restaurant. We want to know how many customers the restaurant gets each day. The current owner tells us. We don't know if he's lying or not. So we observe how many come in each day for a week. Our data:

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Why not use a matched pairs test here?

  • $n = 6$, $d̄ = 0$, $s_d = 7.56$
  • $SE = \frac{s_d}{\sqrt{n}} = \frac{7.56}{\sqrt{6}} = 3.09$
  • $t = \frac{d̄ - D_0}{SE} = \frac{0-0}{3.09} = 0$
  • $p = 1$
  • Conclusion: fail to reject the null.

More generally, why would we ever use a Chi square goodness of fit test instead of this matched pairs test? It seems very similar to me.

I know that the Chi square distribution is produced when we square a normal distribution. I have a vague idea that with the matched pairs test we're taking the average of the difference, but with the Chi square goodness of fit test, we're taking the average of the square of the difference, so I suppose it makes sense that the distribution of this statistic will be a Chi square distribution. But why do this? And why is this test associated with categorical data?

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