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Use the formula for the regression line to predict the mileage for vehicle that weighs 4000 pounds:Can we be very confident; moderately confident; not at all confid...

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Use the formula for the regression line to predict the mileage for vehicle that weighs 4000 pounds:Can we be very confident; moderately confident; not at all confident that this prediction dose t0 the actual mileage? Explain:Compare vour answers about your confidence for parts € and f Do vou think they agree? Explain:Explain the trend context of the graph: Does this make sense? Explain.the association strong; moderate wcak?Plug the value =0 into the cquation and find the valuc for y- That is t

Use the formula for the regression line to predict the mileage for vehicle that weighs 4000 pounds: Can we be very confident; moderately confident; not at all confident that this prediction dose t0 the actual mileage? Explain: Compare vour answers about your confidence for parts € and f Do vou think they agree? Explain: Explain the trend context of the graph: Does this make sense? Explain. the association strong; moderate wcak? Plug the value =0 into the cquation and find the valuc for y- That is the "v- intercept" Find and interpret the value of the y-Intercept here by writing sentence using the names and units of the variables and this Vaiuc Use this sentence: "For . vehicle that weighs pounds, our model predicts that the will be (number) (units) (variable] (number) (units] Does the y-intercept make sense the context of this problem? Explain: Note: This doesn't mean the model = wrongl The model requires an anchor" value for the explanatony variable of zero, which often doesn"t fit inside the valid interval (Think carefully about the equation and ser whether you can tell wthout doing any computation ) What charges predicted value of mPB wnen YOU change from 4000 - 4D01 pounds? Do you have compute values to find this? callthe number multiplied by the explanatory variable (x] the slope; The interpretation of the slope When the value of x increases by unit, then the predicted value of y changes bv Wnits Rewrite this sentence using the numerical value of the slope, the names af tha two variables; and the units of the two variables:



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Please do the following. (a) Draw a scatter diagram displaying the data. (b) Verify the given sums $\Sigma x, \Sigma y, \Sigma x^{2}, \Sigma y^{2},$ and $\Sigma x y$ and the value of the sample correlation coefficient $r$ (c) Find $\bar{x}, \bar{y}, a,$ and $b .$ Then find the equation of the least-squares line $\hat{y}=a+b x$ (d) Graph the least-squares line on your scatter diagram. Be sure to use the point $(\bar{x}, \bar{y})$ as one of the points on the line. (e) Interpretation Find the value of the coefficient of determination $r^{2} .$ What percentage of the variation in $y$ can be explained by the corresponding variation in $x$ and the least-squares line? What percentage is unexplained? Answers may vary slightly due to rounding. Miles per Gallon Do heavier cars really use more gasoline? Suppose a car is chosen at random. Let $x$ be the weight of the car (in hundreds of pounds), and let $y$ be the miles per gallon (mpg). The following information is based on data taken from Consumer Reports (Vol. $62,$ No. 4 ). Complete parts (a) through (e), given $\Sigma x=299, \Sigma y=167, \Sigma x^{2}=11,887$ $\Sigma y^{2}=3773, \Sigma x y=5814,$ and $r \approx-0.946$ (f) Suppose a car weighs $x=38$ (hundred pounds). What does the least-squares line forecast for $y=$ miles per gallon?

Alrighty guys, let's go ahead and jump into a problem 19 here. So I'm gonna go and solve the entire problem using a google sheet. Um So using a google sheet it does take a little longer upfront, but as time goes on, it definitely does save time in the long run. So let's go ahead and input all of our values here. Mhm. Almost done. Mhm. Area. Okay, so all those values are now in. Put it. All right, so now we can go ahead and start doing the problem. All right, so I'm gonna go and highlight all of my data and I'm gonna go insert chart where to go. There is cool. And there is my scattered chart there. Okay, so I definitely do want to scatter chart. Um if it doesn't have a scatter chart, it's not a huge deal. Um you know, it doesn't give you a scatter chart right away, It's not a huge deal, you can just make one yourself. Um But I'm gonna make it a little smaller because I do need a little bit more space here. Um There we go. Perfect, okay, cool, so that is my scattered chart but it does look like they want us to do the trend line. So if I click on. So uh but if I click on these three dots, I can do the trend line. So just go to edit chart, this was up earlier, but I just put it back up now. Um If you don't get a scatter chart. No worries. Just click on the chart type and then select scatter chart. If you don't have it already. Also make sure before we do the trend line that these two are checked as well. Okay the next thing we're going to do to get our trend line is we're gonna go customize series and then we're going to scroll and click trendline and they're right. There is our trend line but we do need the equation. So let's go ahead and find the equation really quickly. You're going to scroll down farther and go label use equation and there is our equation. Okay cool so it's a very very very small trend line there. So negative six point 17 E. To the negative three um is going to be um very small there. So that means we need to move the decimal place over three times to the left and so yeah I'll show you how to do that just a second. All right. Um But yeah that is our equation right there. Um So uh interpret the slope and intercept appropriate. Okay sounds good. So um the slow the slope means for what is the increase or decrease in our Y. Value for every increase of one and the X value. Okay so basically what it's asking is this uh or basically what that means is this if we increase the car by £1? Very small increase the car by £1 we expect the mpg. We expect the mpg to decrease by 10.0 negative 0.0 617. And if you can imagine just moving the decimal place over three times. So there's two zeros between the decimal place and the six. Of course that's negative. So as we increase, so the slope would be as we increase the car by £1, we predict. This slope predicts that the MPG will decrease by total of 0.00617. So again, very small number. Uh the y intercept, I don't think is appropriate here because the y intercept is when the X value is zero, is it possible to have a car of weight zero? Well no that doesn't make any sense at all. So we went ahead, we interpret the slope but in terms of the y intercept in this case is not appropriate. So we'll go and skip that one. Um It wants us to predict now it wants us to predict the mpg of the mustang and do the residual. Okay, so the mustang was this Garrett here? So this is our mustang another by looking at the text. So let's go ahead and do a prediction And let's go ahead and use our equation. So negative 0.00617. That asterisk, I see their means times and we do need to multiply by the rate and then we add the white intercept and it looks like it's a predicted value of a little over 21 21 MPG. Now the residual is we just do observed minus predicted and so um yeah, so it looks like, let's see um We're going to do 21 139 -20 and that's our individual and it looks like we overestimated. Um it looks like we overestimated there. So it looks like it's supposed to be 21.139 but in actuality it's 20 so it looks like it's below average there. Okay, so that's below average because we think the average supposed to be about 21.139 and um yeah and so yeah, it looks like it's a little below there. Um So yeah, that actually boots actually got I'm so sorry actually got this order wrong. It should be it's observed. So um this minus predicted that. Ok, that's how it is. So yeah, it's gonna be that one. I'm so sorry about that. Okay. And so then yeah that is that's a residual because it's it's below right there and um yeah, so that we went ahead so that is the ford mustang is below average based on weight. Um party wants us to draw this line we just did and then um and then let's see would it be reasonable and part e the the answer um is probably no. Okay. Um Yeah the answer is probably not, it would not be reasonable to do so why because all of these that we're observing right here are gas cars. Okay, so, um all these here are gas cars, which means that um yeah, which means that it's not the similar things. So you want the data set to be consistent, the same type of car. So it would not be reasonable to do that because because yeah, I think the data would be skewed there because it's a different kind of car and it might give us um, an inaccurate picture of these type of cars with MpG.

All right, we are going back to a little bit of review whether you're doing regression equations. I've used my online graphing calculator. Does most feel free to check it out. But I basically plugged in X values. Why values? And it can spit out of linear regression equation for you. And so that equation is and you can kind of see the n value there. Why equals negative 0.16 if I run to the nearest 100th X cause 24,700 22 point and actually looked up the bba here because around it to the nearest 100 that's gonna be 1.26 There's a regression equation. Find the our value, Will. They already did that for us. It is. You can see it there. Let me just highlight it. R is negative 0.970 Run it to three decimal places. This is a strong correlation answering part C like Is this accord good predictor Bad court predictor. It is a strong predictor. So it is a good predictor. Um, we say anything where the our value or I should say the absolute value of the our value is greater than 0.75 It is a good predictor. You can also see it's really close to all the points, all right? And so Part D 19,000 is that good for 60,000 bucks? Well, it is definitely definitely knocked. And here's why. If our X values air all over my alleges here, if you plug in 60,000 into this equation, I actually went ahead and found what output you get. And so let me just copy down our equation. And if you plug it in, you actually get the output of I haven't written down here 14,000 $972 and 63 cents. So according to the equation, it should be just below $15,000. Another way to say this is Hey, this car that has 30,000 miles is priced at 19,000. So why is one with twice as many mileage twice as many e miles? I should say, price, that the same thing. This is overpriced for this reason here and that reason there

Part one. The Poland L s estimate of beta one oh is 0.36 zero. If the change in concentrate concentration is 0.1, then the change in the log of fair would be Beijing one head times the change in concentration and that would be 0.36 times 0.1, which is 0.36 That implies airfare is estimated to be about 3.6% higher. Part two, The 95% confidence interval obtain using the usual L s standard error is 0.301 2.419 And if we use the fully robust standard Iran's we will get point 245 and 2450.475 which is wider than the one above. The wider confidence interval is appropriate as the neglected serial correlation introduced uncertainty into our parameter estimation. Yeah, Part three. The quadratic has a use shape form, and the turning point is calculated by mhm taking partial derivative of lock of airfare with respect to lock of distance. And you will set that derivative equal zero. You wouldn't be able to find the value of lack of distance where the slope becomes positive, sir. the value of a lot of distance at the turning point is you will take 0.902 divided by two times 20.103 and you can get 4.38 This is the lock of distance, sir. When you convert it back, the value of distance is exponential of 4.38 Okay, about 80. And the shortest distance in the sample is 95 miles. So the turning point is outside the range of the data, which is a good thing in this case, what is being captured in an increasing elasticity affair with respect your distance As fare increases hard for the random effect, estimate of data one is 10.209 which is a bit smaller than the parent LS estimate. This estimate still implies a positive relationship between fair and concentration. The estimate is also very significant, with a T statistic of 7.88 Part five. The fixed effects estimate of beta one is 10.169 which is lower but not so different from the random effect estimate. And this is so because the value of, um, a perimeter in Equation 11 equation 14.11. Yeah, let's say it's, um, Fate. A hat. The Fed ahead is about 0.9, so random effects and fixed effects as meats are fairly similar. Remember, random effect uses a quasi demeaning. That depends on the estimate of this fada, I suggest in equation 14.11. Hard six. Heterogeneous effect. A supply could capture two types of factors that might correlate with concentration. Variable mhm. First, it could be factors about cities mhm near the two airports, for example, population, education level and type of employers. These factors could affect the demand for air travel, and the second set of factors could be factors relate you geographical features and infrastructure condition, such as highway qualities and whether the city locates near a river. So these factors are able to change over time. But in a short time period, let's say, um, the length of the time study in their sample. They are roughly time constant course, Yeah, and so they are able to be captured by a sub I. There are various factors like that, and it's better if we are able to control for them. So in part seven, it is more appropriate to choose to fix effect, estimate

So for this problem, what we're given is, and as our sample size, that's 3 11. We're going to let x one br displacement X two will be class. Mid size X three will be class Large x four will be fuel premium x five will be. Actually, it's not x five. Why is going to be fuel efficiency? And that's going to be calculated by a highway mpg. So now we have our regression equation, and we first have to determine the necessary sons. So you look at the sum of X I and that's 10 38. Then you look at the sum of X, I squared, and that's 3833.68 Then we look at the sum of why I and that's 80 36. And the sum of why I squared is 212,638 than the sum of X. I Y II gives us 2 25 432 7. So now we want to determine the slope of B um, and that's accomplished by B equals and times the sum of X y, minus the sum of X times the sum of why all over end times the sum of X squared minus the sum of acts square. This is going to end up giving us approximately negative 28825 So the mean is the sum of all the values divided by the number of values. So we're going to get that X mean is 3.3145 and the mean of why is 25 points 8392 The estimate of a of the intercept Alpha is going to be the average of wide decreased by the product of the estimate of the slope. So this is going to look like a is equal to the mean of Roy minus B times the medevacs, which is gonna give us 35.3933 Um, so we can replace Alpha or a with Alpha and be with beta. So now that's going to give us this value right here, which will now be 35 933 minus 2.8825 x one and then for part B, we will use. We can use excel to generate the multiple in your aggression model, and it will give us an output. Since this can't be done on Excel, Uh, we get a bunch of values, but if we put in the correct values will end up getting the correct output. So now we can move on to purchase E um, or actually, with part B, we can do part of us. So once you get the numbers in Excel, we see why hat is equal to be not. Plus B one x one plus b two, x two plus B three x three. Um, so we see that be not is going to equal 29 0074 b one is negative. 16581 B two is 4.4860 b three is 1.8190 and with that we end up getting as a result that are y is equal to 29. 0074 minus 1.6581 x one plus four point 44860 x two Move this for our last term, which is B three x three. That's gonna be plus one point 8190 x three. That's our final answer for beam and then foresee. We have our significance level Alpha. That's gonna be a 0.5 And the given claim is that b I or beta? I rather equals zero. So the null hypothesis states that even claim that the slope is zero. Um, So what we have now is that a church not equals. Uh, actually, h hot is given as b I equals zero and h A is given as b I. There's not equal zero. So the p values in this case we would end up getting abated. To corresponds to P equals zero, and beta three will correspond to p equaling zero point. Um, it's 1234 12345678 zeros 704 It's a very close to zero, but not quite. So if the P value is less than the significance than we will reject the null hypothesis hypothesis. So in this case, he is, um, less than zero and b two case, it's less than zero. So we reject the null hypothesis, which is h not. And in this case, um, it's less than zero is that we reject the significance is too Well, so you reject the null hypothesis again. Then, for part D, we do another excel generation of multiple linear regression model. Um, so we can't do anything with the sell side of things, but we have another one of these equations where it's be not, plus B one x one plus B two, x two plus B three, x three plus B four x four and then with all of our values what we end up getting as our estimated regression equation. It's going to be 29 7123 minus 1.6383 x one plus 3.9984 x two +16700 x three minus 1.585 x four In this case, the significance level. This is the last part. E. Alpha is going to equal 0.5 again. Um, and we and then all hypothesis we'll have data. One is equal to beta two, which equals beta three jiggles beta four, which is zero. And our H sub A is going to, um, ST that at least one of the FBI's is non zero. Besides, for some hi then the P values, uh, corresponding we want to look at, so we C p equals zero and P is less than zero. So we reject we reject h not because the P value is less than the significance. So I reject each nut are null hypothesis.


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