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# Achieving Linearity

By Barbara Peterson,2014-07-08 22:42
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Achieving Linearity

CH. 4 Transformations to Achieve Linearity AP Statistics

Example. The table shows the temperature of an instrument measured as its distance from a heat

source is varied.

a) Describe a general pattern that you notice about the data. Distance Temperature

(cm) (F)

1 130

2 105

3 95

4 87 b) Identify the explanatory and response variables. 5 83

6 80 7 78 8 77 c) Enter data into List 1 and List 2. Make a scatterplot and

sketch it below.

d) Describe the nature of the graph.

e) Calculate linear regression and record the following:

a;___________

b;___________

y;___________

r;______

2r;_____

f) Regardless of the values in e), is a line the best model for this data? ________

g) Let’s check the residual plot. Is the LSRL a good model for this data? ___________

h) Make List 3 = log (List 1)

i) Make List 4 = log (List 2)

j) Make a new scatterplot using List 3 and List 4.

k) Describe the nature of this new scatterplot. How does its form compare to the original?

l) Find the predicted temperature for a distance of 4.2 cm.

J Windmuller, DRHS 1

CH. 4 Transformations to Achieve Linearity AP Statistics 1. Cell phones, a recent innovation, have become increasingly popular with all segments of our society. According to the Strategist Group, the number of cellular and personal communications systems subscribers in the United States have increased dramatically since 1990, as shown in the following table.

No.of Subscribers

Year (millions) #1

1990 5.3

1991 7.6

1992 11.0

1993 16.0

1994 24.1

1995 33.8

1996 43.4

a. Sketch the original graph on grid #1.

Label the axes.

b. Apply a test to show that the cellular

systems are increasing exponentially.

c. Calculate the logarithms of the y-values

and extend the table above to show the

transformed data.

#2

d. Plot the transformed data on grid #2

Label the axes completely.

d. You want to construct a model to predict

cell phone growth in the near future.

Perform linear regression on the

equation.

e. What is the correlation for the

transformed data?

f. Now transform your linear equation back to obtain a model for the original data. Write the

equation for this model.

g. The Strategist Group predicts 70.8 million subscribers in 1998, and 99.2 million in the year

2000. How many cellular subscribers does your model predict for these years?

J Windmuller, DRHS 2

CH. 4 Transformations to Achieve Linearity AP Statistics 2. Suppose that two-variable data has been plotted and that the points show a clearly curved

pattern. In this situation, several methods can be used to transform the data. In each of the

following, data have been transformed to obtain a good model. In each case, what would

be the equation that best fits the untransformed data? (i.e. perform the inverse on each)

(a) ln c = 0.105 d + 0.01

(b) log y = 7.43 + 2.49 log x

3. A study of the fuel economy for various automobiles plotted the fuel consumption (in liters of

gasoline used per 100 kilometers traveled) vs. speed (in kilometers per hour). A least

squares line was fit to the data. Here is the residual plot from this least squares fit.

What does the pattern of the residuals tell you

(a) The evidence is inconclusive.

(b) The residual plot confirms the

linearity of the fuel economy data.

(c) The residual plot does not confirm the

linearity of the data.

(d) The residual plot clearly contradicts the

linearity of the data.

(e) none of the above

4. A set of y-values is transformed to achieve linearity by taking the

lnynatural logarithm of each value. The regression of on x is then

computed as

lny=-3.1 + 2.5x. What is the predicted value of y when x = 3.

A. 25118.9

B. 81.45

C. 4.4

D. 19.36

E. 7.55

5. If a scatterplot of bivariate data produces a curve, which might be appropriate to use to achieve linearity?

A. ln of both variables

B. ln of the response variable

C. log of both variables

D. none would be appropriate

E. all would be appropriate

J Windmuller, DRHS 3

CH. 4 Transformations to Achieve Linearity AP Statistics

6. The productivity of American agriculture has grown rapidly due to improved technology (crop

varieties, fertilizers, mechanization). Here are data on the output per hour of labor on American

farms. The variable is an “index number” that gives productivity as a percent of the 1967 level.

Year Productivity Year Productivity

1940 21 1965 91

1945 27 1970 113

1950 35 1975 137

1955 47 1980 166

1960 67 1985 217

a. Show that the productivity increases exponentially by finding the ratio of any four consecutive y values. Show your ratios.

c. residual plot

b. Make a scatterplot of this data on your calculator then describe the nature of the graph:

c. Make a residual plot of this original data and sketch

on the plot to the right.

d. Plot the log y on x on your TI then sketch d. scatterplot

it on the grid to the right.

e. Record the regression equation for the least squares

regression line of the transformed data.

yf. Perform the inverse and record the equation for :

g. Predict the productivity index number for the year 1958.

J Windmuller, DRHS 4

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