# Scatter Plot And Exponential Model by ClassOf1

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Sub: Statistics                                                                           Topic: Regression

Scatter Plot And Exponential Model

Question :
Annual world crude oil production, 1880 – 1998 (millions of barrels)

Year         Mbbl            Year           Mbbl            Year            Mbbl

1880         30              1945           2595            1976            20188

1890         77              1950           3803            1978            21922

1900         149             1955           5626            1980            21722

1905         215             1960           7674            1982            19411

1910         328             1962           8882            1984            19837

1915         432             1964           10310           1986            20246

1920         689             1966           12016           1988            21338

1925         1069            1968           14104           1990            22100

1930         1412            1970           16690           1992            22028

1935         1655            1972           18584           1994            22234

1940         2150            1974           20389           1996            23380

1998            24441

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*The Homework solutions from Classof1 are intended to help students understand the approach to solving the problem and not for
submitting the same in lieu of their academic submissions for grades.
Sub: Statistics                                                                           Topic: Regression
The data table gives information on the world crude production from 1880 to 1998.

(a) Make a scatter plot that displays the relationship between years since 1800 and the crude oil
production. Comments on what you see.
(b) It appears that no single model will describe the relationship between these variables effectively,
Make a scatter plot of the data from 1880 to 1972. (Be sure to use years since 1800!) Use
transformation to liberalize the data, and then perform an inverse transformation to obtain either
a power or exponential model that describe the relationship. Justify your choice of transformation
with appropriate statistical evidence.
(c) Use your model from (b) to predict crude oil production in 1974. How comfortable do you feel
with this prediction? Support your answer with appropriate statistical evidence.
(d) Now make a scatter plot of the data from 1982 to 1998 (again, use years since 1800). Find a linear,
power, or exponential model that describes the crude oil production during this time period.
(e) Use your model from (d) to predict crude oil production in 2002. how comfortable do you feel
with this prediction? Support your answer with appropriate statistical evidence.

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*The Homework solutions from Classof1 are intended to help students understand the approach to solving the problem and not for
submitting the same in lieu of their academic submissions for grades.
Sub: Statistics                                                                           Topic: Regression

Solution :

(a)     Make a scatter plot that displays the relationship between years since 1800 and the crude oil
production. Comments on what you see.
Solution:

From the scatter plot it seems that there is strong positive relationship between Years and world
crude production.

(b)     It appears that no single model will describe the relationship between these variables
effectively, Make a scatter plot of the data from 1880 to 1972. (Be sure to use years since 1800!) Use
transformation to liberalize the data, and then perform an inverse transformation to obtain either a
power or exponential model that describe the relationship. Justify your choice of transformation with
appropriate statistical evidence.

www.classof1.com/homework-help/statistics
*The Homework solutions from Classof1 are intended to help students understand the approach to solving the problem and not for
submitting the same in lieu of their academic submissions for grades.
Sub: Statistics                                                                           Topic: Regression
Solution:

scatter plot of the data from 1880 to 1972

We use ln (natural logarithm) transformation to liberalize the data we get following scatter plot:

Regression Analysis: ln(y) versus Year_1
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*The Homework solutions from Classof1 are intended to help students understand the approach to solving the problem and not for
submitting the same in lieu of their academic submissions for grades.
Sub: Statistics                                                                                 Topic: Regression

The regression equation is

ln(y) = - 121 + 0.0665 Year_1

Predictor                 Coef                       SEcoef                    T                      p

Constant                  -121.314                   1.985                     -61.11                 0.000

Year_1                    0.066498                   0.001025                  64.86                  0.000

S = 0.1301         R-Sq = 99.6%            R-Sq(adj) = 99.5%

Analysis of Variance

Source            DF           SS          MS        F     P

Regression            1     71.213          71.213 4207.11 0.000

Residual Error 19              0.322         0.017

Total            20       71.534

Unusual Observations

Obs       Year_1       ln(y)         Fit     SE Fit Residual St Resid

1       1880      3.4012        3.7022          0.0643     -0.3010        -2.66R

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*The Homework solutions from Classof1 are intended to help students understand the approach to solving the problem and not for
submitting the same in lieu of their academic submissions for grades.
Sub: Statistics                                                                            Topic: Regression
8     1925      6.9745       6.6946       0.0307       0.2798          2.21R

R denotes an observation with a large standardized residual

From the above output we get fitted regression equation is,

Ln (Y) = - 121 + 0.0665 x.

Therefore y = exp {-121 + 0.0665x}

From the above out put we can see that R-sq = 99.6% this implies that 99.6% of variation in “Y” that is
explained by using the least-squares line and the “X” variable. The first table in the output gives the
estimated coefficients, b0 and b1, and a t-value that test whether the null hypothesis of the
coefficient is equal to zero and the corresponding p-value is given. P-values that are used to test
whether the constant and slope are equal to zero here we can see that p-value for the Year is 0.000.
These indicate that there is sufficient evidence that the coefficients are not zero. This indicates that
the log transformation work well for the given data.

(c)      Use your model from (b) to predict crude oil production in 1974. How comfortable do you feel

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