Math 1 · Unit 10 · Lesson 4 · part 2 of 4
Adds or multiplies? The gnats
Adds or multiplies, even when the data is messy.
Adds or multiplies? The gnats
Adds or multiplies, even when the data is messy. In Lesson 3 a line fit the gnats with r of about 0.98, and the residual plot still showed a curve. The gnats are in the wrong family. When data multiplies instead of adds, fit an exponential: y = a · bx.
Run the check from Units 7 and 9, knowing that real data will only be roughly constant, close to the same, not exact.
| hours | gnats | difference (subtract) | ratio (divide) |
|---|---|---|---|
| 0 | 12 | ||
| 1 | 20 | 8 | 1.67 |
| 2 | 35 | 15 | 1.75 |
| 3 | 60 | 25 | 1.71 |
| 4 | 80 | 20 | 1.33 |
The differences are 8, 15, 25, 20: nowhere near equal, so it does not add. The ratios are about 1.7, 1.75, 1.7, and 1.3: not exact, but roughly the same. And the scatter plot bends upward. That is enough to try an exponential.
Exponential regression finds the exponential curve that fits the data best. Technology does it and gives a and b. Switch between the two fits and watch which one follows the points.
Read the model
y = 12.48 · 1.63x
a = 12.48: about 12 gnats at the start, the initial value.
b = 1.63: the growth factor. 1.63 = 1 + 0.63, so the number grows about 63% each hour.
When will there be 200?
No solving tool for the exponent yet, so use a table or the graph with a horizontal line at 200.
| hours | 4 | 5 | 5.5 | 5.7 | 6 |
|---|---|---|---|---|---|
| model | 88.1 | 143.6 | 183.3 | 202.2 | 234.1 |
About 5.7 hours. That is extrapolation, past the 4 hours of data, so say about.