Problem solving and data analysis
Data inferences — Harder example
- [Instructor] A researcher collecting information about 1,000 randomly selected physical therapists concluded that the median hourly wage for physical therapists in the United States at the time of the study was between $22.76 and $59.24 with a 95% confidence level. Which of the following could represent the median hourly wage, based on the same sample, same sample, for physical therapists in the United States with a 90% confidence level? So let's just think about what confidence level means. That means, remember, the median is gonna be some number, it might be the actual the median hourly wage for physical therapists. It might be $30 an hour, $25 an hour. They're trying to estimate it by doing this random selection and then they're providing a range and they're saying, hey, there's some confidence level that this range captures that true median hourly wage. So when they say there's a 95% confidence level, they're saying that there's a 95% probability that the true median is between these two numbers. Now if we're talking about a 90% confidence level, if we're talking about a 90% confidence level that means we are less confident that the true median is between these two numbers. In order to be less confident, you would wanna have even a narrower range. You would want a range that is a subset of this range right over here. Let me make this clear. So the range is $22.76 all the way to $59.24. So they say it's a 95% confidence level. That means, I'm gonna actually draw a number line here, so let's say these are just points on the number line, these are points on the number line. So there's 22.76, this is 59.24, they're 95% confident that the true median that the true median is going to be that it's going to be between these two values. So they're 95% confident that the true median is there that the true median is there or that the true median is there but there's still a 5% chance that maybe the true media could be here. It could be below this range or above this range. Now if we're talking about a lower confidence level, 90% confidence level, that means that the reign, this range should be narrowed. If you're gonna be less confident, that means you wanna have, or the only way you're gonna be less confident is if you have a narrower range. If you had a broader range, if it went from say here to here, you'd be even more confident. This type of a range you might have a 97% confidence level. So at 90% less confident, you're looking at something that might look something like that might look something like that. Now let's look at the choices. So this is $17 to 64.90, so this is actually more like this one. You're starting lower and you're ending higher. So you should be even more if you're 95% confident that this range capture you should be even more confident that this range captured it. So this would actually maybe be a 97 or 98 who knows confidence level. Not a 90% confidence level so we can rule that out. 20.48 to 53.32. So this one's interesting because it starts lower but then it ends lower too. So I don't know what you could actually what you could say it's based on the same I don't know, this one's a little bit it depends kind of what the distribution that you selected was, they didn't tell us a lot about that. This is a little bit, this one feels a little bit shady. 21.56 to 56.12, this is also similar it starts a little bit lower and then it ends a little bit lower. So it's kind of shifted the range and so this is also, you don't know for sure that you're going to have, I mean you're probably going to have a well, they don't tell us a lot about the distribution so I'm not gonna make too many statements there. Now this last one goes from 25.65 to 56.35. So that's going to be something like that's going to be, actually, like from here to here. This is going to be a narrower range. So you would be less confident. So this could be something that represents a 90%, a 90% confidence level. So actually I would go with this one because this is kind of a purely this is a subset of the previous range so I like this choice right over here.