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Conditions for confidence interval for a proportion worked examples

UNC‑4 (EU)
UNC‑4.B (LO)
UNC‑4.B.1 (EK)
UNC‑4.B.2 (EK)

Video transcript

- [Instructor] Ali is in charge of the dinner menu for his senior prom, and he wants to use a one-sample Z interval to estimate what proportion of seniors would order a vegetarian option. He randomly selects 30 of the 150 total seniors and finds that seven of those sampled would order the vegetarian option. Which conditions for constructing this confidence interval did Ali's sample meet? So, pause this video, and you can select more than one of these. Alright now, let's work through this together. So one thing that you might be wondering is, well, what is a one-sample Z interval? Well, you could really interpret that as he's gonna take one sample and then construct a confidence interval based on that. The reason why it might be called a Z interval is the whole idea behind a confidence interval is you're going to pick a number of standard deviations above and below the true parameter that you are actually trying to estimate, and then use that to make your inferences. And one way of thinking about the number of standard deviations, people will often call that a Z score, or Z is often used as a variable for the number of standard deviations above or below something. So really, he's just trying to construct a confidence interval, but remember, in order to construct a confidence interval, we have to make some assumptions. He's taking, there's 150 students, right over here. He's finding it impractical to survey all 150 to figure out the true population proportion. So instead, he samples 30 of the seniors. So, N is equal to 30. And from that, he calculates a sample proportion. Looks like seven out of the 30 are, they want the vegetarian option. And he's going to determine some confidence level and then construct a confidence interval. But remember the conditions that we've talked about in the previous videos. The first thing is, we have to be confident that, is this a random sample? So that would be the random condition, and that's what choice A is telling us. The data is a random sample from the population of interest. Do we know that? Well, it tells us in the passage here, he randomly selects 30 of the total seniors. So I guess we'll take their word for it. We don't know his methodology of what he considers random, but we'll take their word for it, that yes, this has been met. The data is a random sample. If it said he sampled the football team, well, that would not have been a random sample. The next condition here looks all mathematical, but this is really the normal condition. And the idea behind the normal condition is that, in order to construct these confidence intervals, we're assuming that the sampling distribution of the sample proportions is roughly normal, and it is not skewed to the right or skewed to the left like this. And so, right here it says, look, the sample size times our sample proportion has to be greater than/equal to 10. Or our sample size times one minus our sample proportion has to be greater than/equal to 10. Well, another way to think about this is, our successes in our sample need to be greater than/equal to 10, and our failures need to be greater than/equal to 10. Well, how many successes were there? There were seven. And you could even say, look, our N is 30 times our sample proportion is seven over 30, which is going to be seven. So our successes is less than 10, so actually, we violate the normal condition. Once again, this is a rule of thumb, but this is telling us that our actual sampling distribution might be skewed. Remember, this is just based on one sample, what we're able to figure out. This is one sample Z interval. We might be wrong, but we wouldn't feel good that we're meeting the normal condition here, so I would rule this one out. Individual observations can be considered independent. Well, if he randomly selected people with replacement, then they could be independent. Or, if the people he is selecting, if his sample size is less than 10% of the total population, then it could be considered independent, even though it wouldn't be perfectly independent. But we see here that he sampled 30 people out of 150. So his sample size was 30 out of 150, which is the same thing as one fifth of the population, which is the same thing as 20%. And since this is greater than 10%, we are violating the independence condition. We could have met the independence condition if he was sampling with replacement, which it doesn't seem like he is, or if this thing right over here was less than 10%. But we're not meeting that, so we cannot feel good about that constraint. And so, since we're not meeting two of the three constraints for, I would say, valid confidence intervals, or confidence intervals we would feel confident in, this is not so good of an analysis on Ali's part.