Once again. Now, welcome to a new problem. This time you're dealing with confidence intervals. And whenever you make predictions in statistics, you, uh, dealing with confidence Interval. So what you're saying is that you could either have estimation could either have estimation or hypothesis testing hypothesis testing You could have now an alternative hypothesis. When it comes to estimation, you could use confidence Interval, where you're either say 90% confident that something is gonna happen or you're gonna make a certain prediction or your 95% confident or your 99% confidence. So these are the things that are gonna happen when it comes to, uh, confidence intervals. And your goal in making predictions is, for example, to see that the mean off a sample which you call X. But this is the mean of a sample, uh, predicts the main off the population with a certain level of confidence off course, your errors allowed to happen eso your expo will have some kind of margin off era when you're running a, uh, confidence interval. And so the purpose off building on interval is to say, if you mean is a certain value, you're gonna have some intervals that capture the mean and then you're also gonna have some intervals that don't capture the mean. But if you're 95% confidence, which is the default, uh, Interval, you'll find that off. All these intervals, 95 out of 100 of them are going to capture the mean intervals are not just for making these kinds off predictions. You could also have ah, problem like the case we have where we're dealing with gas prices, uh, gas prices say in, uh, 2003, we have gas prices and you're doing a comparative analysis between California and Florida when it comes to gas prices. And in California, your X bar for God's surprises, which is the mean of example, is two point awful. And then, for the case of Florida, the gas is a little bit cheaper. The overall average price of gas is when you take a sample is a dollar a dollar 72. So it's a dollar 72. Remember, you're using the same sample size in this case on Don't forget that when it comes to the data, uh, the sample size for California is a little bit bigger than the sample size of Florida, which is 35 on then. Off course, we do have population. Standard deviation for California is Tencent's on then population standard deviation off Florida is, uh, eight sense. And this is pretty much what we're saying is that, uh, when you're computing your margin off era, you're gonna need the standard deviations. And we're lucky enough to, uh Well, actually, you're gonna need both the sample size and the standard deviation. So the first requirement, in part A was saying way, Want to get the point, estimate, point estimate? Um, um, that differentiates that differentiates, uh, the population, I mean, gas gallon prizes between these two states. Between these are two states, you know, that's that's what you're looking for. You want to get the differences in gas prices between these two states, and then the other thing you're saying is in part B, you want to get to 95% confidence on the margin off era for making the point estimate prediction for the interval. So margin off error in part c, uh, was saying eso at at at 95% confident. What's the margin of error? That's part B. And then in part C. And just recall that you're getting the difference between the two. A gas prizes. So Expo one minus expert, too on. Then you build an interval off of that in. Patsy was saying the 95 we want to get the 95% confidence interval that estimates that estimates the difference in I was gonna say average, but I'm just gonna say population means the difference in population means between California and Florida. Well, that's what we're looking at. So we're going to jump right into it. And in part, a, um you one is the, um mean Gus prize a gallon in California. Um, YouTube is through the I mean gas prize, but gallon in Florida. Uh, expel one, as we said, is for theme the sample mean And this is this is population. Just remember that this is population the muse. Other population sample mean a gas prize. A gallon in California on export to is simple. I mean, gas prices, the gallon in Florida. So we're looking at that. And so the difference. You know, if you want to compute the difference, the point, estimate difference. Uh, the point estimate a difference becomes mm. the point. Estimate difference for me one minus mu tube is simply Expo one minus expert too, Which is the same was $2.4 minus $1.72. And that gives 0.32 dollars. That's the difference. And then, uh, in part B but be we want to get the margin of error. Three. The confidence interval or the confidence level are the confidence level, uh, is given by 95% confidence. So if it's an interval, you're saying this middle part is 95%. So this dale right here is gonna be Alfa over two, and this deal will also be off over to remember Alfa is the era, which is 20.5 So, um, since we do have standard deviations for the population and we our goal is to get the margin off error. Imagine off era mhm. We like calling it E. That's the sama's Z off over to eso. The disease caused Z off over to this is the Z score tied up to you have a positive Cisco on the left on the negatives isco on the right. So negative there was a critical values that define the computational tool for the margin off error. So Sigma squared one over N one or sigma squid to over into If you go to the table, if you look up the table Z table with an offer off Uh huh. 0.5 and then off over two off 0.5 divided by two, which is 20.4 to 5. You get to see that the Z off over two is 1.96 ZF over two is 1.96 So this becomes a computational aspect with the radical 0.10 squared over 40 plus 0.35 squared off 35. And so our margin off era E is gonna be point all four if you round it. So this is rounded. So we're rounding off the numbers on, then finally, in part C. We want to compute the actual confidence interval, uh, confidence interval for me one minus mu tube, which is the difference in the means. So to articulate the confidence interval, we take the point estimate off the difference between the mean gasoline prices or gas prizes between the difference between me mean gasoline prices between Florida and California so x by one minus expert, too. Now, this is plus minus the margin off era. So Expo One, we already got that Expo one in expert to that was 0.3 to plus minus point. +04 If you do the math on the loss side, you're going to subtract the imagine off era, and then on the upper side, you're gonna add the margin of error, okay? And that gives us a confidence level. So the 95 percent confidence interval estimate, um, full the difference. The difference in mm population mean prizes or gal, um, off gas between uh huh between California and Florida will be equivalent to 0.28 and 0.36 So we're saying what, 95% confidence that the average difference between gas prices for these two states is gonna fall in between this interval. And of course, some of the numbers around it s oh, you know, just take heart from that. So once again, we had a problem. Our goal was to determine the confidence interval off the difference in means between California and Florida. When comes to gas prices. We had, uh, ex BA for California and Florida, and the sample sizes and the standard deviations off population were given. But three requirements get the difference in terms. Off point estimates get the margin of error and then get the confidence interval The difference. 0.32 confidence level. Margin off error We took the multiplier, zesco, multiplier and the standard error. This part is the standard error portion off the two groups and this is the our multiplier, which happens to be 1.96 and that you have to look up in the table so rounded the margin of error 0.4 and then the confidence interval includes Thebe point estimate off difference in means and the, uh, margin off air on both sides. So I hope you enjoy the problem. Feel free to send any questions or comments and have a wonderful day