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The Chi-square test is a statistical method used to assess whether observed frequencies differ significantly from expected frequencies. In this context, it's applied as a **goodness-of-fit** test. This means it checks how well the observed data from the borer groups fits a Poisson distribution model. To perform this test, you calculate the Chi-square statistic using the formula: \[\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}\]Here,

• \(O_i\) represents the observed frequency for each category of borers.
• \(E_i\) is the expected frequency under the Poisson model.

Working through this process allows us to quantify the difference between our observed counts and what we'd expect if the Poisson model accurately describes our data. After calculating the \(\chi^2\) value, it is compared to a critical value from the Chi-square distribution table with corresponding degrees of freedom. This comparison helps decide if any deviations from the expected values could be attributed simply to random chance.

When we talk about the **goodness-of-fit** for a distribution, we are essentially asking: "How well does our model fit the data?" In this exercise, a Poisson distribution is used as a model. It predicts the frequency of events (like the number of borers per group) by first determining the average rate at which they occur.

The goal is to see if the observed frequencies of borers matches what's expected according to the Poisson distribution. Here’s how it’s typically done:

• Calculate the expected frequencies for each category based on the Poisson model.
• Use these expected frequencies to perform a Chi-square test.

Once both observed and expected frequencies are compared, we gauge whether the differences are just random variation. A close match would indicate the Poisson distribution is a good fit; substantial differences would suggest our model might not be appropriate, signaling that a different distribution could describe the data better.

Expected frequencies are crucial when performing a Chi-square goodness-of-fit test. They represent the number of occurrences we'd anticipate for each category, assuming our model is correct. In the case of the Poisson distribution, these are computed using the known mean \((\lambda)\) of the data.First, calculate \(\lambda\), the average number of borers:
\[\lambda = \frac{252}{120} = 2.1\]With \(\lambda = 2.1\), the probability for each category \((k)\) is found using:
\[P(X = k) = \frac{e^{-2.1} \times 2.1^k}{k!}\]From these probabilities, multiply by the total sample size (120) to determine each expected frequency. For instance, if the probability \(P(X = 0)\) is calculated to be a certain value, multiplying by 120 gives the expected number of groups with zero borers.Finally, these computed expected frequencies are directly compared to the actual observed frequencies to assess the goodness-of-fit of the Poisson model. The comparison reveals whether the observed distribution aligns well with theoretical predictions.