The table at the right shows the amount of carbon dioxide in the atmosphere for selected years. Make a scatter plot of the data, letting x be the number of years after 1900. Determine the kind of polynomial model that seems to be most appropriate. Predict the amount of carbon dioxide in the atmosphere in 1993 . Are you confident in the prediction?
year co2 in atmosphere
1968 324.43
1983 342.27
1998 367.07
2003 375.74
2008 384.16
To make a scatter plot, we will plot the number of years after 1900 (x-axis) against the amount of carbon dioxide in the atmosphere (y-axis).
Year (x) | CO2 in Atmosphere (y)
---------------------------------
1968 | 324.43
1983 | 342.27
1998 | 367.07
2003 | 375.74
2008 | 384.16
Scatter Plot:
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400 |
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350 | *
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300 | *
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0 50 100 150 200
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Based on the scatter plot, it appears that the data roughly follows an upward trend, but it does not seem to fit a straight line model very well. The points appear to be curving upwards, suggesting that a polynomial model may be more appropriate.
To determine the kind of polynomial model that is most appropriate, we can try fitting different polynomial models and see which one fits the data best based on visual inspection and the R-squared value. Let's try fitting a quadratic model (polynomial with degree 2).
Quadratic model: y = ax^2 + bx + c
Using the given data, we can create a system of equations to solve for the coefficients a, b, and c.
For the points (x=0, y=324.43):
324.43 = 0a + 0b + c --> c = 324.43
For the points (x=15, y=342.27):
342.27 = (15^2)a + 15b + 324.43
For the points (x=30, y=367.07):
367.07 = (30^2)a + 30b + 324.43
For the points (x=35, y=375.74):
375.74 = (35^2)a + 35b + 324.43
For the points (x=40, y=384.16):
384.16 = (40^2)a + 40b + 324.43
Solving this system of equations will give us the coefficients for the quadratic model.
After solving the system of equations, we obtain:
a ≈ 0.000
b ≈ -0.040
c ≈ 324.430
Therefore, the quadratic model equation is:
y ≈ 0.000x^2 - 0.040x + 324.430
To predict the amount of carbon dioxide in the atmosphere in 1993, we need to find the corresponding x-value (number of years after 1900).
1993 - 1900 = 93.
Plugging x = 93 into the quadratic model equation, we get:
y ≈ 0.000(93^2) - 0.040(93) + 324.430
y ≈ 324.430
The predicted amount of carbon dioxide in the atmosphere in 1993 is approximately 324.430.
However, it's important to note that the quadratic model is based on a limited amount of data and may not accurately represent the underlying trend. Therefore, confidence in the prediction for 1993 is relatively low.