Chi-Square Goodness-of-Fit Tests for Receipt Distributions

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What is a chi-square goodness-of-fit test?

A test that assesses whether observed counts in categories differ significantly from the counts expected under a proposed distribution.

How are expected counts calculated for a category?

Multiply the sample size by the probability that the proposed distribution assigns to that category: $E_i=np_i$.

What is the exponential rate estimate used in the lab?

The reciprocal of the sample mean, $\hat{\lambda}=1/\bar{x}$. The estimated rate determines the exponential model used to calculate percentile boundaries and probabilities.

How do you find an exponential percentile boundary?

For cumulative probability $p$ and rate $\lambda$, use $x_p=-\ln(1-p)/\lambda$.

What does failing to reject the null hypothesis mean?

The data do not provide sufficient evidence against the proposed distribution at the chosen significance level. It does not prove that the distribution is true.

Quiz questions

1. Which expression is the chi-square goodness-of-fit statistic for observed counts $O_i$ and expected counts $E_i$?Answer: $\sum_{i=1}^{k}\frac{(O_i-E_i)^2}{E_i}$
2. In the general chi-square goodness-of-fit procedure, what degrees-of-freedom formula accounts for $k$ categories and $m$ estimated parameters?Answer: $df=k-1-m$
3. For a chi-square goodness-of-fit test, what probability does the p-value represent?Answer: Under the null model, the probability of a statistic at least as large as the observed one.
4. What expected-count condition is generally used for the chi-square approximation to be reliable?Answer: Each expected count should generally be at least 5.
5. If goodness-of-fit tests fail to reject both a uniform model and an exponential model for the same data, what can be concluded?Answer: The data may be consistent with both models; neither is established as uniquely correct.