To find expected frequencies for Chi-Square tests, you need to know whether you’re doing a Goodness of Fit test or an Independence test. Let’s break it down:

1. Goodness of Fit:

   • First, figure out the total number of observations. We can call this total $N$.
   • Next, you need to find out the expected proportion for each category. This might come from some theory or earlier data.
   • Now, to get the expected frequency for each category, use this formula:
     $E_i = N \times p_i$
     Here, $E_i$ is the expected frequency for category $i$, and $p_i$ is the expected proportion.

2. Independence Tests:

   • Start with a table that shows how different categories interact. This is called a contingency table.
   • To find the expected frequency for each cell in the table, use this formula:
     $E_{ij} = \frac{(row \ total \ of \ i) \times (column \ total \ of \ j)}{N}$
   • This calculation helps you see what frequencies you would expect if the variables were independent from each other.

Just fill in your numbers, and you’re good to go!