I’ll give feedback on your work below, but I want to start with noticing that you used a 2-sample confidence interval to answer the question. That is a totally valid strategy for a situation like this, but only because the alternative hypothesis was “different”; had the scenario asked “higher” or “lower” the confidence interval would not work in the same way. Typically, when given a significance level, and asked if there is “convincing statistical evidence” of something, we should be running a hypothesis test. That said, you will still be scored for your work with the confidence interval. The scoring for a “convincing statistical evidence…” scenario includes:
- Stating null/alternative hypotheses
- Defining the parameters in the null/alternative hypotheses
- Choosing an appropriate test/interval by name
- Checking the conditions to run the chosen test/interval
- Writing the results from the chosen test/interval
- Correctly interpreting the results from the chosen test/interval in terms of whether we do or don’t have evidence for the alternative hypothesis.
Given that list (some parts are scored together to create a question with 3-4 scoring components), you can likely see that your work doesn’t have enough there to be earning much of the available credit. You calculate the appropriate margin of error, and therefore obtain a confidence interval, but never name the interval, check conditions (random samples, approximately normal sampling distribution [at least 10 successes/at least 10 failures], 10% condition), or write hypotheses. Additionally, you used “0.05” in the interval, instead of using (0.838 - 0.708 = 0.13) as your difference of proportions to add/subtract the margin of error. That would have led you to a different confidence interval where 0 was not included. Given that your interval did include 0 though, your conclusion that we do not have convincing evidence would get scored as correct, because you interpreted the answer you got correctly. Unfortunately, you would not get credit for the other components of the question.
Whew, that was a lot of writing! I hope I was able to be clear in my explanations though - it’s clear that you know many of the concepts, but need some work in how to address the types of questions the AP exam gives. Having said all of the above, I should note that with the formatting changes of this year’s test, you will likely not need to do the level of calculations that you did. It’s much more likely that you will be given an interval (or p-value, or something else) and be asked to just do the interpretation. Or perhaps choose an appropriate procedure and check conditions, but not actually finish building the interval/obtaining the p-value.
Thanks for reading, and I hope this helps!