Conditions for inference for difference of means | AP Statistics | Khan Academy

A food scientist wants to estimate the difference between the mean weights of eggs classified as jumbo and large. They plan on taking a sample of each type of egg to construct a two-sample t-interval. Which of the following are conditions for this type of interval?

So before I even look at these choices, and they say choose all answers that apply, so it might be more than one. Let's just think about what the conditions for inference for this type of interval actually are.

We've done this many times in many different contexts, and so we first of all have the random condition. That’s the idea that each of our samples is random or that we are conducting some type of an experiment where we randomly assign folks to one group or eggs, in this case, to one of two groups.

In this case, we are taking two samples, and we would hope that they are truly random samples. The second is the normal condition, and the normal condition is a little bit different depending on whether we're talking about means or whether we're talking about proportions.

The random condition is essentially the same. The normal condition, when we're talking about means—remember, they're looking at the difference between mean weights of eggs—means you would want your data to meet certain criteria. There are actually several ways to meet the normal condition. One is if the underlying distribution is normal.

The second way is if your sample sizes for each of your samples are greater than or equal to 30. So if your first sample size is greater than or equal to 30 and your second sample size is greater than or equal to 30. Or even if the underlying data—you don't know whether it's normal or if it isn’t normal—and even if you aren't able to meet these, as long as your sample data is roughly symmetric and not skewed heavily in one way or the other, then that also roughly meets the normal condition when we're dealing with means.

The third condition, and we see this whether we're dealing with means or proportions or differences of means or differences of proportions, is the independence condition. This is the idea that either your individual observations are done with replacement in both of your samples, or that the sample size for both of your samples is no more than 10 percent of the population. Then you have met this condition.

So that’s a little bit of a review. Let's see which of these apply.

They observe at least 10 heavy eggs and 10 light eggs in each sample, so this actually is the normal condition when we are dealing with proportions, not for means. So I would rule this out; it's a good distractor choice.

The eggs in each sample are randomly selected from their population—yep, that's the random condition right over there. So I would select that.

They sample an equal number of each type of egg. This is a common misconception that whether we're dealing with means or proportions, when we're thinking about the difference between, say, means or the difference between proportions, that somehow your sample sizes have to be the exact same. That is not the case; your sample sizes do not have to be the exact same.

So we would rule this out as well. So right over here, they have listed the random condition. They could have also listed the normal condition and the independence condition.