Data Transformation Activity
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AP STATISTICS | Transforming data using the natural log | SHUBLEKA
Activity 1: Transforming data using the natural log of X. 1. The table below gives the age (in years) and the diameter (in inches) for twenty-seven oak trees. Age Diameter Age Diameter 4 0.8 23 4.7
2. 3. 4. 5. 6. 7. 8.
5 8
0.8 1
25 28
6.5 6
8 8
2 3
29 30
4.5 6
10 10
2 3.5
30 33
7 8
12
4.9
34
6.5
13
3.5
35
7
14
2.5
38
5
16
4.5
38
7
18
4.6
40
7.5
20
5.5
42
7.5
22
5.8
Look at a scatter plot of the data. Analyze the regression model. Take the natural log of the age, L1, and store in L3. Graph the scatter plot (ln(age), diameter) (L3, L2). Find the linear regression for L3, L2. Graph the regression equation with the scatter plot. Analyze the regression model of the form y = a + bx
9. Rewrite the equation to the form y = f ( x). Remember: x = ln(age) , so use substitution to write your equation logarithmic form. 10. Enter this equation in Y2. 11. Graph Y2 on the scatter plot L1, L2. 12. How well is the data captured by this line? 13. Use the TRACE function to predict the diameter of an oak tree that is 24 years old. 14. Find the logarithmic regression for L1, L2. 15. How do the answers compare? 16. Note the values of r and r2 in the linear regression of the transformed data and in the logarithmic regression of the original data. 17. How well does the line fit the data? 18. Plot the residuals. 19. How are the residuals grouped?
AP STATISTICS | Transforming data using the natural log | SHUBLEKA
Activity 2: Transforming Data using the Natural Log of Y The table below contains the length (in inches) and the weight (in pounds) of twenty-five alligators captured in Florida: Length 94
Weight 130
Length 88
Weight 70
74 147
51 640
72 74
61 54
58 86
28 80
61 90
44 106
94 63
110 33
89 68
84 39
86
90
76
42
69
36
114
197
72
38
90
102
128
366
78
57
85
84
82
80
86
83
Source: Burrill, et al. Modeling with Logarithms, 1999 Repeat the steps in the first activity using the given data and by transforming WEIGHT so that L3 = LN (WEIGHT). Run exponential regression before the comparison step. Summarize what you have learnt in these activities in two short paragraphs. Related Mathematics: 1. Exponential Regression is of the form ln y = a + bx or y = e a +bx = e a ebx Substituting
e a = A, eb = B we get y = AB x as the TI-83 Plus shows.
2. Logarithmic Regression is of the form: y = a + b ln x or y = a + ln x b 3. Power Regression is of the form: ln y = a + b ln x
a = ln c and we get: ln y = ln c + b ln x
For some c,
ln y = ln c + ln x b ln y = ln cx b y = cx b With a = c this becomes y = axb as the TI-83 Plus shows.
Activity 1: Transforming data using the natural log of X. 1. The table below gives the age (in years) and the diameter (in inches) for twenty-seven oak trees. Age Diameter Age Diameter 4 0.8 23 4.7
2. 3. 4. 5. 6. 7. 8.
5 8
0.8 1
25 28
6.5 6
8 8
2 3
29 30
4.5 6
10 10
2 3.5
30 33
7 8
12
4.9
34
6.5
13
3.5
35
7
14
2.5
38
5
16
4.5
38
7
18
4.6
40
7.5
20
5.5
42
7.5
22
5.8
Look at a scatter plot of the data. Analyze the regression model. Take the natural log of the age, L1, and store in L3. Graph the scatter plot (ln(age), diameter) (L3, L2). Find the linear regression for L3, L2. Graph the regression equation with the scatter plot. Analyze the regression model of the form y = a + bx
9. Rewrite the equation to the form y = f ( x). Remember: x = ln(age) , so use substitution to write your equation logarithmic form. 10. Enter this equation in Y2. 11. Graph Y2 on the scatter plot L1, L2. 12. How well is the data captured by this line? 13. Use the TRACE function to predict the diameter of an oak tree that is 24 years old. 14. Find the logarithmic regression for L1, L2. 15. How do the answers compare? 16. Note the values of r and r2 in the linear regression of the transformed data and in the logarithmic regression of the original data. 17. How well does the line fit the data? 18. Plot the residuals. 19. How are the residuals grouped?
AP STATISTICS | Transforming data using the natural log | SHUBLEKA
Activity 2: Transforming Data using the Natural Log of Y The table below contains the length (in inches) and the weight (in pounds) of twenty-five alligators captured in Florida: Length 94
Weight 130
Length 88
Weight 70
74 147
51 640
72 74
61 54
58 86
28 80
61 90
44 106
94 63
110 33
89 68
84 39
86
90
76
42
69
36
114
197
72
38
90
102
128
366
78
57
85
84
82
80
86
83
Source: Burrill, et al. Modeling with Logarithms, 1999 Repeat the steps in the first activity using the given data and by transforming WEIGHT so that L3 = LN (WEIGHT). Run exponential regression before the comparison step. Summarize what you have learnt in these activities in two short paragraphs. Related Mathematics: 1. Exponential Regression is of the form ln y = a + bx or y = e a +bx = e a ebx Substituting
e a = A, eb = B we get y = AB x as the TI-83 Plus shows.
2. Logarithmic Regression is of the form: y = a + b ln x or y = a + ln x b 3. Power Regression is of the form: ln y = a + b ln x
a = ln c and we get: ln y = ln c + b ln x
For some c,
ln y = ln c + ln x b ln y = ln cx b y = cx b With a = c this becomes y = axb as the TI-83 Plus shows.
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