I probably don't look at it the right way but I also get the majority of these type of questions right. If you have Kaplan 2015 LN, my text below has some references.
According to Dr. Stanley Smith Stevens, who came up with this idea that there are four levels of measurement: nominal, ordinal, interval, and ratio. I use two of them to answer these type of questions. Chi-square, is like nominal data. Nominal data is things that are counted in groups or categories (gender - male/female, language - english/benagli). The chi-square is essentially testing to see if two nominal variables are independent. You can do this with any number of groups 2x2, 2x3, 3x3. So if you are asked as an example: What statistical test would you run to test whether there was a difference between the cumulative mortality rate for Denver (28%) and Fort Collins in 1985 (14%)? It is chi-square. The idea is since the mortality rate of Denver and Fort Collins are INDEPENDENT of each other (they have to be since they are completely different cities in different locations in the country) we use chi-square to measure the mortality rate. This is the categorical outcome, not a mean value of mortality rate.
If you do a t-test/ANOVA, it is like Interval data. Interval data is things that are measured by a dimension graded in equal intervals (thermometer - Olymayakon, Russia coldest place on earth/my future BP during step 1). If you want to be technical ratio data is like temperature because it has an absolute zero, but not relevant for the purposes of this discussion. So when you look at the means of two interval data and compare them that is when you want to do a t-test (2+ means)/ANOVA (3+ means).
