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I am trying to follow instructions to figure out the linear regression and quadratic regression. Here is the data:

 

A water bottle is filled with water. The water is allowed to drain from a hole made near the bottom of the bottle. The table shows the water level, y in centimeters, from the bottom of the bottle after x seconds.

 

Time (seconds) 0 20 40 60 80 100 120 140 160 180 200 220
Water Level (cm) 42.6 40.7 38.9 37.2 35.8 34.3 33.3 32.3 31.5 30.8 30.4 30.1

 

1. What is the linear regression equation?

 

2. What is the quadratic regression equation?

 

3. Which is a better fit?

 

4. What is the approximate water height after 240 seconds?

 

5. How might choosing the incorrect regression equation affect predictions?

Guest Oct 15, 2017
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3+0 Answers

 #1
avatar+26329 
+1

This graph might give you a start:

 

.

Alan  Oct 15, 2017
 #2
avatar+1373 
+1

You're in luck; I have a TI84 Plus CE, and I know how to do this.

 

1. Press Stat and Enter

 

After doing this, you should see a screen that looks like the following:

 

L1L2L3L4L5...
------------------------- 
      

 

2. Input The Time Data into Land Water Level into L2

 

After doing this step, your screen should now look like this:

 

L1L2L3L4L5...
042.6    
2040.7    
4038.9    
......    
22030.1    

 

3.  Press 2nd+Stat Plot+Enter+Enter

 

Be sure to press the keys, in order, from left to right. By doing this, you have enabled the first stat plot. Make sure that the type is the one that is a scatter plot before proceeding. For me, this is done by default, so there are no change. You'll notice that Xlist and Ylist have already defaulted to L1 and L2, which is good. If you see all of this, you can proceed to the next step.

 

4. Press zoom+9

 

You should now see a graph with your data points inputted. It should something like the graph below. 

 

 

5. Press 2nd+mode

 

Doing this will quit out of your current window and return you to the home screen.

 

6. Press 2nd+0+x-1+Down Directional Arrow 12 Times

 

2nd+0 opens the catalog where all functions on the calculator are present. Pressing x^-1 will bring you to the D-functions where functions that start with a D, in this case DiagnosticOn, are present. By following the instructions above, there should be an arrow (it looks like ►) pointing towards DiagnosticOn. Once you see that, press enter twice. Your homescreen should now show Done.

 

This step is crucial. You may think you can skip this step because it is insignificant, but it becomes relevant later on.

 

7. Press Stat+Right Directional Arrow+4

 

Your screen should look like the following:

 

              LinReg(ax+b)

Xlist:L1

Ylist:L2

FreqList:

Store RegEQ:

Calculate

 

Make sure that your cursor is blinking directly to the right of Store RegEQ before going to the net step. Use the directional arrows, if necessary.

 

8. Press vars+Right Directional Arrow+Enter+Enter

 

Doing this will now show a Ynext to Store RegEQ. 

 

9. Down Directional Arrow+Enter

 

The calculator will now calculate the line of best fit. You should see the following screen:
 

      LinReg

y=ax+b

a=-.057...

b=41.111...

r=.951...

r2=.975...

 

Write down the r2 number rounded to the nearest thousandth place because it will come handy later. Also, if you were to skip step 6, you would not see the r. This is why turning on the diagnostic is necessary.

 

10. Press Stat+Right Directional Arrow+5

 

The screen should look exactly the same as step 7, but there should be QuadReg at the top now. This is where the quadratic regression will be calculated. Scroll down to Store RegEQ: again.

 

11. Press vars+Right Directional Arrow+Enter+2

 

Doing this will display a Y2 next to Store Reg EQ: . 

 

12. Press Down Directional Arrow+Enter

 

You should now see this:
 

   QuadReg

y=ax^2+bx+c

a=2.103...E-4

b=-.103...

c=42.654...

R2=0.999...

 

Record the R^2 value from the quadratic regression as well.

 

13. Press Graph

 

You should now see the scatterplot, the line of best fit, and the quadratic of best fit. It should look something like this:
 

 

You have done all the work needed with the calculator. You can put it down now.

 

1) Rounded to the nearest thousandths place, the linear regression is \(y=-0.057x+41.111\)

2) Rounded to the nearest thousandths place, the quadratic regression is \(y=\left(2.104*10^{-4}\right)x^2-0.103x+42.654\)

 

3) I hope you recorded those r^2 numbers because we are going to put them to use. Whichever \(|r^2|\) is closest to 1 is the regression equation we should use. Since the r^2 for the quadratic is closer to 1, we should use that one. 

 

4) Ok, remember when I said to put down the calculator? I was wrong. You need the calculator for this question, too. 

 

1. Press 2nd+trace+enter+240

 

The calculator will calculate the expected water level after 240 seconds. However, you'll notice that it calculated it for the linear regression. We'll fix that in the next step.

 

2. Press the Up Directional Arrow

 

You should see an x- and y-coordinates at the bottom. It will say:

 

x=240           y=29.9477...

 

Therefore, the water level will be approximately at 29.948cm

 

5) Choosing the incorrect regression type may result in a skewed understanding of how something may behave in the future.

TheXSquaredFactor  Oct 15, 2017
 #3
avatar+26329 
+1

Nicely detailed TheXSquaredFactor,  though I think the use of R^2 here is, perhaps, overkill!

 

Also, if part 5 refers specifically to the two fits here, I would expect to see the word "underpredict" appear.

.

Alan  Oct 15, 2017

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