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Chapter 13, Section 13.1, Problem 021fThe following table gives information on the amount of sugar (in grams) and the calorie count in one serving of sample of [3 v...

Question

Chapter 13, Section 13.1, Problem 021fThe following table gives information on the amount of sugar (in grams) and the calorie count in one serving of sample of [3 varieties of Kellogg's cereal:Sugar (grams)20Calories110 200 140 T10 T00 30 190 100 120 190 T90 100 120The predictive regression equation of the number of calories on the amount of sugar is ; = 84.694 + 5.408x, where x is amount of sugar (in grams) and Y is calories_ Estimate the calorie count for cereal with 58 grams of sugar per

Chapter 13, Section 13.1, Problem 021f The following table gives information on the amount of sugar (in grams) and the calorie count in one serving of sample of [3 varieties of Kellogg's cereal: Sugar (grams) 20 Calories 110 200 140 T10 T00 30 190 100 120 190 T90 100 120 The predictive regression equation of the number of calories on the amount of sugar is ; = 84.694 + 5.408x, where x is amount of sugar (in grams) and Y is calories_ Estimate the calorie count for cereal with 58 grams of sugar per serving Round your answer to the nearest integer. calories Comment on this finding Little confidence should be placed in this estimate. Much confidence should be placed in this estimate



Answers

The author collected data from 16 cereals consisting of the sugar contents (in grams per gram of cereal) and the calories (per gram of cereal). STATDISK was used to find that the linear correlation coefficient is $r=0.765$ and the equation of the regression line is $\hat{y}=3.46+1.01 x,$ where $x$ represents the sugar content. Also, the mean calorie amount is 3.76 calories per gram of cereal. What is the best predicted calorie content for a cereal with 0.40 gram of sugar per gram of cereal?

Okay, so to check whether or not there existed linear relationship between the variables calories and sugar, we can use the mini type software to construct a scatter plot with the two variables. So now, from the scatter plot, we can see that there is just no linear relationship between the variables sugar and calories. Because we can't really draw the line of best fit. Because no matter how you draw it, the points are just too scattered to show a relationship. So it does not make sense to construct confidence and prediction drills based on the least square regression equation, because there is a relationship between calories and high protein and sugar content and moderate portion and adventure ship bars. Okay, no, We have to find a 95% confidence interval for the mean sugar content of energy bars. So, using the T test in the many type software, we can yeah, Get the following output for one sample. Yeah. Sugar. Yeah. Uh Yeah. And it's 14, I mean, Switching .71 Standard Deviation 9.3. Mhm. We went to I mean, that's one 95% confidence interval. Hmm. So from the debates have suffered. How put the 95% confidence interval for the mean streak content of energy bars is and 9.4985 to 19.9 301

Alrighty guys. So um to go and start with this problem um looks like with the I'm gonna have to look at the book and the enters at the same time so far apart. It's asking if the calories sugar content are linear related. And the answer is no, mainly because if you look at the data, you have to look on page 213 in your text. Um The data does not appear to follow a linear pattern. I kind of see a lot of dots kind of scrambled all over the place. Doesn't really seem to be a linear pattern. So um No just that's just based on looking at the dots and part B actually confirms that suspicion because the covid determination is only 6.8%. That's pretty low. That means that 6.8% of the variation in sugar, which is the why value by the way Is explained by the least squares regression line and 90 over 93% of the variation is explained by other factors. So it's not a great predictor right there. Um In part C is a little bit uh and then does the support? Yes it does. Because if uh if it did fall in your pattern, that percentage would be much higher than 6.8. It be probably closer in the 90s probably. Whereas in this case 90 is explained by other factors besides the least course aggression lines. So it looks like we're right on part. Um And so for the next one. Um what I want to go ahead and do is this it looks like we have to draw scatter diagram. So what I did is I took the liberty of typing down all the values, You'll find those values on page two and 13 um in the table and I added the last two values that 80 and the six like it says in part C at the very end. And I'm going to go ahead and have google sheets create that scattered diagram for me and so I'm going to go, I think it's insert yes insert chart and there it is and it gives me a scatter chart right there which is great. Um And then yeah, just scatter chart. If it gives you something else besides the scatter chart, you can go and just click on scatter chart, that's fine. Then you're gonna want to go to customize, I'm pretty sure it's series and double check. Yes, it is awesome. Okay, so customized series and then we're going to go a trend line when you click on that and we're going to show R squared and we are going to use the equation and there's our new equation right there. Notice R. R squared and this is actually because confirmed the text is much higher. That means that if we go back to here, since our, since we have that new are square to 42% Are square in this case as your .41 which is 4211%. Um that means that we're supposed to interpret the results of 42.1% of variation and sugar is explained by the least squares regression line and 57.9% is explained by the factors. And then why did that happen though? So why did that have such a big impact? Well, number one, there's a couple of things I see. Number one notice how far away from the data set this lives. This is almost like an outlet here. Okay, so this almost looks like an outlier because it's so far away from the other data points which is going to us could see our day a little bit. So that's number one. Number two, it's also just adding a data point. The more data points you add, the better the regression line is at, the better the regression line is at predicting points. And so we add another point. So r squared went up. And also because it's a bit an outlier which could skew our data. Just a touch

So you have this table and we're going to be making a scatter plot of carbs as the X and energy as the Why. So let's start by doing that. So it looks like our X values are going from about 22 pt. So let's have 20. Ah, fair Any 40 50 60 a 70 and an 80. And our Y values the energy it looks like it's ranging from about 92 little over free 100. So let's just do this. 100 1 50 200 2 50 300. Now all we have to do is plots all of our points. So we have 21.1 comma in 92. Then we have 32 comma 1 40 Now I'm referring to something off screen. Get my points. So 21 x 92. Why Bernie two x 140 y keep going down the list. I just don't want to keep scrolling back and forth, so I'll keep referring to my list off screen. 4100 and 80. Chill, Verity. Ah 120. 45 1 88 41 60 55 to 20 77 is free 20 35 is 1 50 and 75 is to 96. So is there a linear relationship? It's pretty clear the answer is yes. This almost perfectly makes a straight line. So we're going to give this one a yes. Now for the harder step, make the line of best fit. This can take a long time. If you do it by hand, you can make an Excel spreadsheet that will just do all this for you and we'll skip straight to the end. It depends on what Your teacher A wells. You might want to talk to them. If you have to do this by hand because it's gonna take a long time. Let me show you what I did here. They should work in over programs like Excel, too. But I made a calmer the exes and calm for the why and then our sample size And and then over here I made an X squared a Y squared and X times y all I did was put the corresponding box squared same with y square in the corresponding box squared X y just multiply the two corresponding boxes. Then over here is where we get all this stuff that we need some of X. You just do the some of the column with your ex. Some of why do the some of the calm with your wife some of X squared. Take the sum of your X squared column Some of y squared. Take me some of your white column. Some of X y take the sum of your X Y column. Now, to find some of Squared X, you're gonna have to refer back to the equation. So K two that is our some of X squared that minus the sum of X squared. So I just wrote it twice here. So some of X times some of X divided by our sample size. So that is the 10 over here. Sum of squares of why is the same faint except you use the Y values sum of squares of X y. So this one starts with B some of X y over here and to minus the I, too. So some of X times some of why it j two divided by the sample size and then you can get the two values were most interested in Slow is just some of square of X y divided by the sum of square vax. So for me, it's k fine, divided by I five and the y intercept. So we start off with the some of why over J two minus are slope times the sum of X And then this whole fame is divided by the sample size. Then we ever slope and why intercept now, If you do have to do these by hand, I will do some of the work that set it up for you. So, for example, um, some of acts and some of X squared is something that we need it. You would find this by going up to the data set and going to the X, which is carbs to find some of X. You just add all this up to find some of X squared, you take each value, square it and then add all them up. You can't just take the sum of acts and square it because the order of operations will change the answer. So if, for example, you did this by hand, you would need the sum of acts and you would get that That is for 51.7 and you would need the sum of X squared and the sum of X squared would be to free 5 to 8.77 And then we have our sample size. That's 10. So you can now find the sum of squares of X, which would be equal to 23,005 to 8.77 minus. So are some of X for 51.7 squared, and this is over 10 and you would get that he sum of squares of axis free 1 to 5 point for one. Then you would p you do a similar process for sum of squares of X y find your slow and then find your y intercept. And at the end of the problem, you will get that the equation. Sorry. You don't get that. The equation is why equals so our slope, which was free 0.92 times X plus or why intercept, which is 9.55 Let me go back for a minute. I would just get it from here. The Y intercept and the slope and again the equations for why intercept in Sloper right here Em is always our slope and B is always our y intercept, if that wasn't clear. But consult your teacher for what programs you are and aren't allowed to use for these questions Now for the last part. So if you wait 40 in 65 carbs, how much energy would you gain? Well, if you remember, carbs was our X axis. So we're just gonna plug in 40 in 65 for X. So for this one, we're going to get why is equal to free 10.92 times 40 plus 9.55 and that is equal to 166.35 And then you would do the same thing for D. Except with 65 this time, free 650.92 times 65 plus nine point by five. And that is equal to 2 64.35 And we are done.


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