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Use the data shown in the table that shows the number of bacteria present after a certain number of hours. Replace each y-value in the table with its logarithm, logy. Find

the equation of the regression line for the transformed data. Then construct a scatterplot of (x, log y) and sketch the regression line with it. What do you notice?

Number of hours, x
1
2
3
4
5
6
Number of bacteria, y
99
205
428
780
1455
2740
5203

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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)

| *

| *

| *

| *

| *

| *

| *

|*

|_____________________

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).

User Kaushik Shankar
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