Right. This problem is about a pharmaceutical company making tranquilizers, and we are given some data, and the problem is broken up into five different parts, so it's gonna take a little bit to go through it. Uh, the first part, part A is asking us to determine some statistics based on the provided data. So the first thing I'm going to do is I'm gonna bring in my graphing calculator and I'm going to hit this stat button and I'm going to select edit, and I'm going to place all my numbers into the graphing calculators. So I've put all the effective periods into the graphing calculator, and from there I'm going to have the calculator helped me determine all the statistics that I need. So in part, a subsection one, you're asked to determine the mean and to find the mean we're going to total up all the pieces of data, and we're going to divide by how many pieces of data there are. And when I total up those values, I get 22.6. I divide by the fact that there's nine pieces of data and I get approximately 2.51 now. Another way I could have done it because my calculator has the data I could hit, Stat. I can scoot over to calculate I could select one variable statistics and tell the calculator where you have stored your data and we have stored it. Enlist one. And by doing so, you can see now that the average is listed right there. X bar is 2.51 or thereabouts for part B of part two of this, it's asking us to determine the standard deviation, and it's denoted by S a backs. And we can get that right from our graphing calculator, which is recorded right here. And as I round that to three decimal places, it's approximately 30.318 in part three of part. Hey, we need to determine the value of N and N represents the sample size and the sample size. There was nine pieces of data, so it's gonna be nine. And in part four off part A. It's asking you to determine end minus 11 lower than nine would be eight, and there's a reason we're doing the end minus one, and it's going to be found when we do Part five of this problem gonna get rid of the calculator for the time being and let's go to Part B in Part B. It's asking us to define the random variable X inwards. So what does the X stand for? And in this particular problem, the X represents the effective length of time in hours, mhm for a tranquilizer. So how long does it last before it wears off? And then, in part C, we need to define X bar? And in this particular problem, it's the mean, effective length of time again. That's in hours for a tranquilizer from a sample of nine patients because there were nine pieces of data. So that stands to reason that there were nine patients. Part D in This is asking you which distribution should be used to solve this problem. So for Part D, we're going to use the students T distribution to solve this. And the reason we're going to use that is because we do not know the population standard deviation. We know the standard deviation of our sample. We calculated that using the graphing calculator, but we don't know the population standard deviation. And when we don't do that or We don't know that we use our T distribution. Now it's on to Part E and in part E. It comes in three parts and it's asking you to determine the confidence interval. It's asking you to sketch a graph, and it's asking you to calculate the error bound and in order to solve these, we can't solve them in the order that they're asked. What we do need to do is solve part three. First, we've got to find the error bound and the error bound of a mean using A T distribution will be solved using the formula T sub Alfa over to multiplied by s over the square root of n now S and n were defined in part A to find that t value and it's called the critical T value. We're gonna have to think about the confidence interval that we're going to determine. So we are trying to construct a 95% confidence interval for this population. Mean so the best approach is to draw out a bell curve and think about the fact that 95% of the bell is in that center, so that leaves 5% unaccounted for. That's out in the two tails. So the Alfa in this problem is 0.5. So therefore the Alfa over two is going to be 20.5 divided by two or 20.25 So what we're saying is we have 2.5% of the bell in the right tail and 2.5% or to five as a proportion in the left tail. Now, to calculate the tea that we're going to use in this formula, we're going to utilize our graphing calculator and we're going to use the feature in the graphing calculator called University. And when we use Unversity then asks for the probability or the area followed by the degrees of freedom. So let me take you to our graphing calculator and to access the university, you're gonna hit the second button and the variables button and select number four. Now, the probability that we're going to put in is what we put in those tails, which was 0.25 and the degrees of freedom we find by doing that end minus one that we answered in part A. So are degrees of freedom is going to be nine, minus one or eight. So we're putting in verse T we're putting our area that's in the tail, and then we're using eight Syrian type in our eight, and we get in answer a negative 2.306 So that value is corresponding to the T score associated with the location. That's to the left of the center of the bell. So we've got negative to negative 2306 here, and the one on the right side is going to be positive. 2306 And when we're using the formula and again, the formula is right here, that formula E B m error bound of mean equals, we're only going to use the value, the absolute value of those answers. So we're just going to use the 2.306 the S from part A was the 0.318 and the end from Part A was a nine serving divide by a square root of nine. And we're going to get the answer for the error bound of the mean, which is the part three of a letter E to be 30.2387 Well, now we're gonna go back and we're going to calculate the confidence interval. So now we're gonna go back and solve part one of part E. Now the confidence interval. What is the confidence interval? If we think of it on a number line, we have our estimate of the average here. And what we're going to do is we're going to add the error. We're going to subtract the error and it's then going to give us a new interval. So in interval format, we're trying to find a confidence interval at the 95 percentile. So we're going to take the average and we're going to subtract the error and then we're going to take the average and we're gonna add the error. So the average in this problem was two point five. And let's just go back up and do a double check. It was 2.51 So we're going to subtract the error that we got, which was this 0.2387 and then we're going to take the average. We're gonna add the 0.2387 And when you do that, you're going to get to 2713 as a low bound of our confidence interval, and we're gonna have 2.7487 as our upper boundary of our interval. So just to recap for part e sub three, our error was 0.2387 And for part one, which was the confidence interval is ah, low bound of 2.2713 and an upper boundary of 2.7487 And then finally, for this problem or this part, we needed to sketch the graph. So now we're going to do part two of Letter E. We're going to sketch the graph. So for part two, you're going to draw a bell shaped curve. We're going to show the 95% in the center we're gonna put the average in the center, which was 2.51 We're gonna put our lower boundary on the left boundary 2.2713 and we're going to put the 2.7487 on the upper boundary. And we're now talking about this confidence interval going from 2.2713 up to 2.7487 with a 2.51 as the average one more part to this problem we need to do part E or sorry, Part F and part f of this problem is asking you What does it mean to be 95% confident in this problem? If we were to sample many groups of nine patients, mhm 95% of those samples when you calculated the confidence interval would capture the true population mean of tranquilizer effective time. And I like to always give a visual of this. What do we mean by that? The visual would be, we have an average and the average was 2.51 and we select nine patients and we construct a confidence interval. And sometimes the confidence interval captures the mean and other times we calculate a confidence interval and the mean is an inside that interval. And what it's saying is, 95% of the time the interval is going to be capturing. That mean there's gonna be 5% of the time that the interval doesn't capture it, and it's to the left or to the right of the mean. So hopefully that helped you with E six parts of this problem