Question

Questions 1 through 6 work with the length of the sidereal year vs. distance from the sun. The table of data is shown below.

Planet	Distance from Sun (in millions of miles)	Years (as a fraction of Earth years)	ln(Dist)	ln(Year)
Mercury	36.19	0.2410	3.5889	-1.4229
Venus	67.63	0.6156	4.2140	-0.4851
Earth	93.50	1.0007	4.5380	0.0007
Mars	142.46	1.8821	4.9591	0.6324
Jupiter	486.46	11.8704	6.1871	2.4741
Saturn	893.38	29.4580	6.7950	3.3830
Uranus	1,794.37	84.0100	7.4924	4.4309
Neptune	2,815.19	164.7800	7.9428	5.1046
Pluto	3,695.95	248.5400	8.2150	5.5156

Enter the original data in L1 and L2 (that is, the Distance from the Sun and Years).

Make L3 = ln(L1) and L4 = ln(L2). Verify that this matches the columns given above. Don't worry about the small discrepancies you may find due to rounding and the number of decimal places shown on your calculator. If your results differ from the values above, double-check your original entries!

1. Draw a scatterplot of Distance vs. Year (using the untransformed data) with the least-squares regression line. Does the line seem to model the relationship well? (2 points)
2. Do a linear regression for the following: Distance vs. ln(Year) (L1 vs. L4, if you entered the data as directed above), Ln(Distance) vs. Year (L3 vs. L2, if you entered the data as directed above), and Ln(Distance) vs. Ln(Year) (L3 vs. L4, if you entered the data as directed above).

   (Note that the explanatory variable is always some form of "Distance.") To get the most out of this Assignment, look at a scatterplot of each of these combinations.

   Which transformation yields the highest correlation coefficient (Pearson's r)? Sketch a scatterplot of this transformation and show the least-squares line. What is the value of r and r² for that transformation, and what regression equation does it yield? (3 points)

3. Using the equation from number 3, make a residual graph. Create a residual plot on your calculator and interpret it; you don't need to draw the plot. (Note: You'll probably need to turn off the plot in Y1 to display the scatterplot correctly.) (2 points)

4. .    Using algebra, convert your regression equation to a power equation (show your work below). Enter this equation in Y2 (press Y= and enter the equation) and make a scatterplot of L1, L2, with Y2, verifying that the power equation is a good fit for this data. Finally, summarize, in plain English, what you've done in questions 1-4. (3 points)

   As you set up your regression equation, keep in mind that the variables are ln9 and lnx .

5. The purpose of the transformations you're studying is to find a simple model to describe the relationship in a data set. The model can be used to predict a response value (called interpolation for values within the range of the data set and extrapolation for values outside the range of the data set). Recall that extrapolation is usually not a valid way to predict y-values. A well-known feature of our solar system is the asteroid belt between Mars and Jupiter. One theory about the asteroid belt is that it's made of primordial material that was prevented from forming another planet by the gravitational pull of Jupiter when the solar system was formed. One of the largest asteroids is 951 Gaspra. Its distance from the Sun is 207.16 million miles. Use your linear regression equation to interpolate the length of its sidereal year. (1 point) Remember that you need to take the natural log of Distance before you plug it in, and that your first result will be the natural log of Year. Show your work.
6. Finally, calculate the length of the year for 951 Gaspra from the power function you developed in Question 4. (Show all your work) (1 point) Note: Theoretically, the answers from 5 and 6 should be the same, but they'll probably come out differently due to rounding between steps. The more digits you carry throughout the calculations, the closer the two answers will be.
7. So Ive answered questions 1-3 already, but dont get 4-6. If anyone could solve them, that would be great! If you can, include work for 1-3 as well so i can double check my answer.

Answer 1