The following is a solution to number one. And this asks about the conditions for inference that must be met whenever you're testing um for the population, mean, you and there are two things to look for. The first thing is that you have independence in your samples independence and the main things you look for there, we need to have random sampling. Okay, So I should say it should say something with a random sample or randomly sampled. And also the sample size needs to be no more than 10% of the population size. So the small end, that's the sample size needs to be less than or equal to 10% of the big end, which is the population. Um that means that each especially without replacement. So whenever you you pull someone from the sample, it doesn't make a huge dent in the population. And the second thing we look for is that the sampling distribution of X bar is approximately normal. And there are two scenarios where that can happen. If it's pulled from a population that is normal, then the sample size can be, you know, whatever. But if the population is not normal, we don't know the distribution of the population then And needs to be at least 30. Okay, so either the population is normal or if it's not normal or we don't know, that sample size needs to be at least 30 and then it asks us when to use the normal model or the Z distribution and when to use the T. Distribution. And basically you're gonna use the z distribution when sigma is known. So if you know what the population standard deviation is, um then you can use the z distribution. But if you don't know what sigma is, then you have to use this t. Distribution. So whenever sigma is unknown, you'll need to use the T distribution.