
Concept explainers
a.
To plot: The
Given information:
The atmospheric concentration of carbon dioxide (in parts per million) recorded by the observatory in years between 1960 and 2015.
x | y |
1960 | 317 |
1970 | 326 |
1980 | 339 |
1990 | 354 |
2000 | 369 |
2005 | 380 |
2010 | 390 |
2015 | 401 |
Plot:
Take the year as x coordinate and concentration as y coordinate, and then draw the scatter plot as shown below:
Interpretation:
From the scatter plot it seems that the linear regression will best fit the scenario.
b.
To find: If a linear or a quadratic model fit more appropriate.
A linear model seems more appropriate for the scatter plot.
Given information:
The atmospheric concentration of carbon dioxide (in parts per million) recorded by the observatory in years between 1960 and 2015.
x | y |
1960 | 317 |
1970 | 326 |
1980 | 339 |
1990 | 354 |
2000 | 369 |
2005 | 380 |
2010 | 390 |
2015 | 401 |
Calculation:
Observe that the points do not fall nearly along a well-known curve. But, they do seem to fall near a line with upward slope.
Since the curve drawn seems to be a straight line, a linear model seems more appropriate for the scatter plot.
c.
To find: Find the regression equation for the appropriate model, and superimpose the graph of the regression equation on the scatter plot.
The regression can line by written as
Given information:
The atmospheric concentration of carbon dioxide (in parts per million) recorded by the observatory in years between 1960 and 2015.
x | y |
1960 | 317 |
1970 | 326 |
1980 | 339 |
1990 | 354 |
2000 | 369 |
2005 | 380 |
2010 | 390 |
2015 | 401 |
Calculation:
Prepare the table to find the regression coefficients as follows;
Use the table to perform the calculations as shown below.
Then,
Therefore,
So, the regression can line by written as
Now, plot the regression line as shown below.
The graph of the regression line best fit the data. That means, the graph shows that the model appears to work well.
d.
To find: The approximate concentration level of carbon dioxide in the year 2025.
The concentration of carbon dioxide in the year 2025 will be
Given information:
The atmospheric concentration of carbon dioxide (in parts per million) recorded by the observatory in years between 1960 and 2015.
x | y |
1960 | 317 |
1970 | 326 |
1980 | 339 |
1990 | 354 |
2000 | 369 |
2005 | 380 |
2010 | 390 |
2015 | 401 |
Calculation:
The model for the given situation is
Substitute
Conclusion:
The concentration of carbon dioxide in the year 2025 will be
e.
To explain: The circumstances that might caused the predication in part (d) of the mark and tell whether the true value will be higher or lower than the predicted value.
Given information:
The atmospheric concentration of carbon dioxide (in parts per million) recorded by the observatory in years between 1960 and 2015.
x | y |
1960 | 317 |
1970 | 326 |
1980 | 339 |
1990 | 354 |
2000 | 369 |
2005 | 380 |
2010 | 390 |
2015 | 401 |
Explanation:
If the model is predicted to be an upward quadratic, then the prediction in part (d) would have been off the mark.
Since an upward quadratic function gives a higher value than a linear function with upward slope, the true value would have been lower than the predicated value.
Chapter 1 Solutions
EBK PRECALCULUS:GRAPHICAL,...-NASTA ED.
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