Answer:
To find the equation of the regression line for the transformed data, we need to replace each y-value in the table with its logarithm, log y.
Using logarithm base 10 (log10), we can calculate the logarithm of each y-value:
log(99) ≈ 1.9956
log(205) ≈ 2.3118
log(428) ≈ 2.6314
log(780) ≈ 2.8921
log(1455) ≈ 3.1628
log(2740) ≈ 3.4378
log(5203) ≈ 3.7166
Now, we have the transformed data:
Number of hours, x
1
2
3
4
5
6
Transformed y-values (log y)
1.9956
2.3118
2.6314
2.8921
3.1628
3.4378
3.7166
To find the equation of the regression line, we can use linear regression analysis. The equation of a linear regression line is given by:
y = mx + b
where m is the slope of the line and b is the y-intercept.
Using statistical software or a calculator, we can calculate the regression line equation for the transformed data:
log y ≈ 0.4317x + 1.5647
Now, let's construct a scatterplot of (x, log y) and sketch the regression line:
```
Scatterplot:
(x-axis: Number of hours, x)
(y-axis: Transformed y-values, log y)
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| *
| *
| *
| *
| *
| *
|*
|_____________________
1 2 3 4 5 6
Regression line: (approximately)
y ≈ 0.4317x + 1.5647
```
In the scatterplot, we can observe that the transformed y-values (log y) roughly follow a linear pattern. The regression line represents the best-fit line that approximates the relationship between the number of hours (x) and the logarithm of the number of bacteria (log y).