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AP Stats: UNC‑4.F (LO), UNC‑4.F.1 (EK), UNC‑4.F.2 (EK), UNC‑4.F.3 (EK), UNC‑4.F.4 (EK)

Video transcript

- [Instructor] We are told that a zookeeper took a random sample of 30 days and observed how much food an elephant ate on each of those days. The sample mean was 350 kilograms, and the sample standard deviation was 25 kilograms. The resulting 90% confidence interval for the mean amount of food was from 341 kilograms to 359 kilograms. Which of the following statements is a correct interpretation of the 90% confidence level? So like always, pause this video and see if you can answer this on your own. So before we even look at these choices, let's just make sure we're reading the statement or interpreting the statement correctly. A zookeeper is trying to figure out what the true expected amount of food an elephant would eat on a day. You could view that as the mean amount of food that an elephant would eat on a day. If you view it as the number, all the possible days as the population, you could view this as the population mean for mean amount of food per day. Now, the zookeeper doesn't know that, and so instead they're trying to estimate it by sampling 30 days. So let's think about it this way. If I... So let's say that this is the true population mean, the true mean amount of food that an elephant will eat in a day. What the zookeeper can try to do is, well, they take a sample. In this case, they took a sample of 30 days. And they calculated a sample statistic, in this case, the sample mean of 350 kilograms. I don't know if it's actually to the right of the true parameter, but just for visualization purposes let's say it is. So let's say sample mean, and this is their first sample, it was 350 kilograms. And then using the sample, they were able to construct a confidence interval from 341 to 359 kilograms. And so the confidence interval, I'll draw it like this. We actually aren't sure if it actually overlaps with the true mean like I'm drawing here, but just for the sake of visualization purposes, let's say that this one happened to. The whole point of a 90% confidence level is if I kept doing this, so this is our first sample and the associated interval with that first sample. And then if I did another sample, let's say this is the mean of that next sample, so that's sample mean two, and I have an associated confidence interval. And that interval, not only the start and end points will change, but the actual width of the interval might change depending on what my sample looks like. What a 90% confidence level means, that if I keep doing this, that 90% of my confidence intervals should overlap with the true parameter, with the true population mean. So, now, with that out of the way, let's see which of these choices are consistent with that interpretation. Choice A, the elephant ate between 341 kilograms and 359 kilograms on 90% of all of the days. No, that is definitely not what is going on here. We're not talking about what's happening on 90% of the days, so let's rule this choice out. There is a 0.9 probability that the true mean amount of food is between 341 kilograms and 359 kilograms. So this one is interesting, and it is a tempting choice, because when we do this one sample, you can kind of say, all right, if I did a bunch of these samples, 90% of them, if we have a 90% confidence interval or 90% confidence level, should overlap with this true mean, with the population parameter. The reason why this is a little bit uncomfortable is it makes the true mean sound almost like a random variable, that it could kind of jump around, and it's the true mean that kind of is either gonna jump into this interval or not jump into this interval, so it causes a little bit of unease. So I'm just gonna put a question mark here. In repeated sampling, okay, I like the way that this is starting. In repeated sampling, this method produces intervals. Yep, that's what it does. Every time you sample, you produce an interval. That capture the population mean in about 90% of samples. Yeah, that's exactly what we're talking about. If we just kept doing this, if we have well-constructed 90% confidence intervals, that if we kept doing this, 90% of these constructed sampled intervals should overlap with the true mean. So I like this choice. But let's just read choice D to rule it out. In repeated sampling, this method produces a sample mean between 341 kilograms and 359 kilograms in about 90% of samples. No, the confidence interval does not put a constraint on that 90% of the time you will have a sample mean between these values. It is not trying to do that. It is definitely choice C.
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