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The ! mean sales price for a certain car dealership is S17,000 with a Standard Deviation of S500. Use the Empirical Rule to determine the followingThe percentage of...

Question

The ! mean sales price for a certain car dealership is S17,000 with a Standard Deviation of S500. Use the Empirical Rule to determine the followingThe percentage of buyers who paid between 516,500 and S18,000 The percentage of buyers who paid S16,000 or lower The price range ol 68% of buyers The percentage of buyers who paid S18,000 or higher

The ! mean sales price for a certain car dealership is S17,000 with a Standard Deviation of S500. Use the Empirical Rule to determine the following The percentage of buyers who paid between 516,500 and S18,000 The percentage of buyers who paid S16,000 or lower The price range ol 68% of buyers The percentage of buyers who paid S18,000 or higher



Answers

A Dealer's Profit The following data represent the profit (in dollars) of a new-car dealer for a random sample of 40 sales. (FIGURE CAN'T COPY) Determine the shape of the distribution of new-car profit by drawing a frequency histogram. Find the mean and median. Which measure of central tendency better describes the profit?

Our mu is equal to 10,192 and our alternative hypothesis would be the opposite of that. So it would say our population means is not equal to 10,192. And also given in this problem is at our sample size, our end equals 50. Now, here's an interesting situation in this problem. We're not given an ex bar or an s. Um, and this problem, we're not going to standard deviation either. So how do we compute these? So for calculating X bar, we follow the this formula. Export equals this sum of all the individual data points divided by the sample size. So we're using the Web file that the book provides us. We can calculate this sum, which is 491,000 500 divided by 50. Now, if you want to use excel for this, you can use the following formula equals in all caps, some and then the range of the values. So for me, it was a one a 12 a 51 okay. And that evaluates to 9750 now to calculate the standard deviation we use the following formula s is equal to the square root of this son of the difference between each individual data point and the sample mean squared over and minus one, and that is equal to 1399.99 If you wanted to do that in excel, you would do S t d e v dot es And then your date arrange a 12 a 51. So now that we have our our S and R x bar, we can compute a T test statistic to find our P value. So to compute a T test statistic that is equal to our sample mean minus our population mean divided by our sample, um, standard deviation divided by the square root of our sample size, which is equal to forth out or 9750 minus 10,192 over our sample standard deviation of 1399.99 divided by the square root of 50. And this evaluates to roughly negative 2.3 are native to 0.23 Okay, so if we draw our t distribution, T equals zero in the middle. This value would lie to the left of T equals zero somewhere here. And we're interested in finding the area to the left of this T equals negative 2.23 This represents that this area represents the probability that tea is less than or equal to negative 2.23 Now, in order to find this probability value, we will need to compute a degrees of freedom. So we take our end. So a degree of freedom is equal to and minus one, which is equal to 49. Now we go to our table, we find where and is equal to or degrees of freedom is equal to 49. We've scanned across that row and we figure out where our test statistic lies. So our test statistic was equal to ah, where is it is equal to negative 2.23 Now, remember that because this this charge asymmetrical from tea from T equals 02 um, the left and it's symmetrical from T equals zero to the right. Uh, this value over here is also equal to the probability that he is greater than or equal to 2.23 It's equal to this value so we can use this chart to find where to 0.23 lies that lies be in between 2.1 and 2.405 So it lies. So our probability lies between 0.25 and 0.1 So I will write this on a new page Probability that tea is less than or equal to negative. Two point 23 is between. Is there a 0.1 and zero point 0 to 5? Make sure those are the values. Yes. Okay. But this is not our p value because we have a Our alternative hypothesis is that mu is not equal to something. So this requires a two tailed test and to tail test. So in order to find a p value, we would multiply this times two. So 0.2 is less than or equal to two times the probability that tea is less than or equal to negative 2.23 which is a sinner equal to 0.5 So our P value is a member of the set. You're a 0.2 You're a point verified R p value lies in this range somewhere. Now, in the in part, C were asked if compare our P value to an Alfa of 0.5 So our P value lies in this range and because the two is less than R P value, which is less than or equal to 0.5 which is less than or equal to 0.5 and this is our Alfa over here, we can reject the no hypothesis. So what does this mean for us? That means that there is sufficient evidence sufficient evidence to claim that the mean used car price at this dealership is different, then the national mean of $10,192.

So here is a picture of the hist, a gram, and if you look at it and also there's the measurements of the mean and the median. Now they are not close together. The closer they are, the more symmetrical the data will be. These are pretty far apart that data on the history graham is skewed, so the better descriptor will be the median. If it was symmetrical, it would have been the mean, but for this one it is the median.

We are going to be evaluating the empirical rule on looking at different approximate percentages of a normal distribution based on some intervals that were given in standard deviations on evaluating the correlations between them. So to start off, I always like to create a visual. Think that's easiest. We're talking about these, so you're mean, Were you? Mu is going right in the middle because again it's an average. Then we're gonna draw out our standard deviations. So 12 and three negative one negative, too and negative three. So let's say we want to see in this normal distribution which again just a reminder it is symmetrical. So we want to see what less than the mean would look like. And what percentage of our data would be less than the mean. So if you wanted to draw a line right down the middle there so you can kind of see and Segway off your bell curve? What percentage would be less than the mean? So that's all this over here, this shaded region, so 100% would be everything, and just that shaded region is 50%. And then, if you want to see what's more than the mean. That would be 50% because again it's symmetrical. So everything is the exact same. So 50% would be under the mean 50% for data would people are. But what do we see when it's greater than one standard deviation above the mean So the normal distribution based on the empirical rule we already know within one standard deviation all the data and their 68% of your data is gonna fall within one standard deviation. So based off of that, we want to see the 100% of the bell curve minus 68% which is what balls in that standard deviation that we shaded in. Okay. And then we're actually going to divide that by two because of the two sides or two different tales, and that should give you 16%. So again, if you want to look at it in a visual standpoint, render almost over your really quick. So we already have this shaded region right here, which 68% falls with them. What we're trying to find is over here and over here. So it's 100% minus the 68 to get these circles from here. But then we need to divide by two. Because we don't really want to find that we want to find greater than so this one right over here, that would be positive. OK, so that's kind of gonna put a visually what we just did with math. Okay, so just get rid of that. Makes it a little cleaner. So let's say we wanted to then find what it would look like for anything less than one standard deviation above the mean. So again, find visuals. Very helpful. Oops. We're trying to find we already know 68% is within one, and we would want to find were prompted to find anything less than one standard deviation above the mean. Okay, so one just secrecy it. So anything less than one standard deviation. So from your mule up upto one standard deviation positive and then all of this over here. Last man. So we have 68% already, but we need to find the additional over here. So you want to look at it in math? We already have the 68% but we need to add what's over here or question mark the rest of that. So it would be the 100% left of the bell curve minus 2 68 So again, little visual over here, we already have, you know, 68 shaded in here out of the whole thing. We're subtracting that out. So, what that would give us Are these right here? Okay. And then you divide it by two. Because we don't care about that one over there. Were trying to find this one right here with less than and that would give you after you add on to the 84 excuse me. 68. It should give you 84. Okay. So just do it really quickly. It would be the 100 minus 68 would be 32 by that by, to which we found at the top will be 16. So it's basically 68. Plus the 16 over here, which would give you 84% is all that over there. Let's shaded. Okay, so we've already been prompted to find those, and then what we're gonna dio the last one was Keep that there in case you need it for reference. The last one we're gonna look at is we want to toe see between one standard deviation below the mean and two standard deviations above the mean, so again visualisation standpoint from you for me just smack in the middle. And then we want to find out between one standard deviation below the mean and two above. So again, let's just write it out to be consistent. 12 and three, negative one negative to negative three. So between one standard deviation below. So that's gonna be over here and then two standard deviations above. So it's all that and there. So, using the empirical rule, we already knew that between long two on three they want to negative three between the first inter deviation 68%. But if we go out to two here, 95% falls within that. So we can use that and look at the 68% so the ones in the middle can shaded 68 the middle right there, shaded one center deviation over, plus 95% minus that 68% and divide it by the two different tales. So, again, that's one standard deviation. We know what two standard deviations we're here and we're just subtracting out the redundancies of the 95 the 68 dividing by two and then re adding on to the 68. And that should give you 81 0.5% would fall within one standard deviation below, so that negative one over here and positive two standard deviations, so 81.5% would fall within that in that original sheeted region.

Alright, this problem is about 84. Used, uh, cars. It's a sample so are in value is going to be 84 and they gave us some more data on the sample. They told us the sample mean was $6425 and they told us the sample standard deviation was $3156. This problem does come in four different parts. And when we get to part, see, there's parts within parts C. So let's tackle part. A party is asking us which distribution should you use for this problem and you should be using the student's T distribution. And the reason we are using the student's T distribution is because we do not know the population standard deviation. We do have a standard deviation that was provided us, but it is referring back to the sample of 84 used cars. So because we do not know the population standard deviation, we will use the student's T distribution for Part B. It's asking us to define the variable X bar inwards, so X bar is going to represent the mean cost of the 84 used cars. Now for part C. There are three parts to it, and I'm going to tackle them a little out of order. So in part C, it's asking us to come up with the confidence interval to sketch the graph and to calculate the error bond are bound in order to come up with the confidence interval. We do have to find the error bound first, so let's tackle that. So we're trying to come up with a confidence interval at the 95% level. And in order to do that, we're using the formula that says Arrow error bound of the mean equals T sub alfa over to multiplied by the standard deviation of the sample divided by the square root event or the square root of the sample size. So because we're trying to find the 95% confidence interval, we're going to put 0.95 in the center of that bell shaped curve. The Alfa, in this case is three remaining part of the bell, and if 95% is accounted for, that means there's 950.5 unaccounted for, so we need to find the T value standard T value associated with Alfa over to so Alfa being 0.25 Alfa over two is going to be a 20.25 And that's referring to how much is in this left tail and the area that's in the right tail. And in order to calculate this t value, we're going to have to use our graphing calculator and we're going to access the inverse T function. And when you use that function, it does ask you for the area of the curve that's in the left tail. So it's gonna be 0.25 And it also asks you for the degrees of freedom, and our degrees of freedom is always going to be n minus one. In this case, we were using 84 um, in our sample. So therefore are degrees of freedom is 83. So I'm gonna bring in the graphing calculator and show you where in verse t is. So you're going to hit the second button, the distribution or variables button and number four. And it's going to ask you what is the probability or the area in the left tail, which is 0 to 5, and it asks you for your degrees of freedom and your degrees of freedom in this case was 83. So therefore our T value is negative. One point, uh 989 So what is that? Referring thio That negative 1.989 is referring to the lower boundary of this confidence interval. And because the T distribution is symmetric, then this upper bound is going to be positive. 1.989 So, in our formula, the error bound of a mean we're going to use the 1.989 for the tea value. Our s value in this problem was 3156 and our sample size was 84. So therefore our error bound is 684.91. So when I said we were going to do the problems out of order, what we have just found is the answer to part three of letters C. So now let's go back and tackle part one off the part C and that's asking us to actually come up with the confidence interval. And it's I always like to dry picture to kind of reference this confidence interval. So what we will do is we're gonna put the average in the center, and we're going to add the error bound for the mean and subtract the error, bound for the mean in order to get the low boundary and the upper bound the boundary of the confidence interval. It's giving us that wiggle room so our confidence interval at the 95% level is going to be X bar, minus the error bound and then x bar, plus the error bound. So in terms of the numbers of this problem, it's gonna be 6000 425 minus 684.91 and 6425 plus 684.91. So that is going to result in a confidence interval of $5740.9 and an upper boundary of $7109.91. So what we have just found is the confidence interval, the 95% confidence interval for this data and it is part one or I of part C eso There's one more part in part C. It asks you to sketch a graph. So we're going to go back to our bell shaped curve and we're going to put the low bound on the left side of the confidence interval. We're going to put the upper boundary on the right side and right in the center, we're going to include our average, which is 6400 25. So this refers to the 95% confidence interval, and that picture is going to be part two of part C in this problem. All right, There's one last part to this problem Part D. And in part D. You are asked to explain what a 95% confidence interval means for this study. And it would mean that if we were to sample n equals 84 many times. So we're gonna grab a sample of 84 cars, then we're going to grab another sample of 84 cars. Then we're gonna grab another sample of 84 cars. We're gonna do that many times than we would expect to see 95% of the confidence intervals that we have generated. They're going to capture the true mean. So again, I like to draw an image to go along with this meaning as well. So the picture that I would draw would look like this. So we've got our true mean and in this case, we really don't know what it iss. All right, we have a sample mean but we don't know the true mean. So our true mean is right here, and it's saying that 95% of the different confidence intervals that we would create would capture it. So maybe we'd have one confidence interval here. And we have another confidence interval here and another confidence interval that spans this with and there's another confidence interval. But spans this myth this with 95% of the confidence intervals going to capture that true mean. But there is going to be 5% of the time that we don't capture it. So, for example, we might end up with the lower boundary being higher than the true mean. So notice this last one this interval here did not capture that true me


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