Ok: two more questions to break down.
Question 3:The goal of this question was to identify the correct methods to compute different probabilities in a situation. Students need to select the correct probability model for computing probabilities: a normal distribution for an individual result, a smart application of the “at least one” rules of probability (or, perhaps apply a binomial model), and a normal model for the sampling distribution of a mean. Topic: depth measurements of the earth.
- Not abiding by the mantra: Your picture is your path. Surprisingly, some students did not identify they were using a normal model (by drawing a picture or mentioning “normal.”
- Students struggled to use a binomial distribution to compute P(“at least one measurement is positive.”) They were more successful if they used the complement rule ( P(at least one success) = 1-P(no successes).
Question 4: The goal of this question was to determine the appropriate inference procedure for a scenario (2-smaple or matched pairs t interval), execute the appropriate mechanics for the interval, and interpret results in correct context. Students also needed to show how to use the interval to make a decision about statistical significance.
Students did not distinguish between a completely randomized design and a matched pairs design. As a result, they picked the wrong test.
Students could not construct the correct formula for the confidence interval of interest. I was surprised how often this happened. This disappointed me, because I provided many tools to help them practice this throughout the year. They have a formula packet, an extra credit review task on this topic, and other tools. At the very least, they could identify the name of the test, and write out the correct inputs. You can omit the formula and get full credit for stating the correct test name (“paired 1 sample t interval, mean difference”) if you write out the inputs to the interval with correct notation/ names. But if you write out the wrong formula, you may get no credit (wrong test, wrong calculations to answer)
Students confuse “interpreting the meaning of the interval in context” with “interpreting the confidence level of the interval.” …And then they get that wrong. Firstly, interpreting the interval simply means you’re trying to get a range of plausible values for the true mean difference – just put context to that idea. A common “Failed attempt:” In all possible samples, the true difference in growth rates will fall between 0.97 cm and 3.01 cm 95% of the time. This is NOT true: We assume, with this procedure, that the true mean difference is fixed. Repeated intervals constructed from new samples of the same size have a 95% success rate at capturing the true mean difference in growth rates.