Question
The equation that best models the linearized data for a particular data set is log y equals negative 0.02396 x plus 4.8153.
What does the number negative 0.02396 tell you about the pattern exhibited by the points in the scatterplot of the original data set?
Based on the equation, what is the value of y when x = 0? Show your work.
What does the number negative 0.02396 tell you about the pattern exhibited by the points in the scatterplot of the original data set?
Based on the equation, what is the value of y when x = 0? Show your work.
Answers
GPT-4o mini
The equation you provided is in the form of a linear equation for the logarithm of \( y \):
\[
\log y = -0.02396 x + 4.8153
\]
### Interpretation of the Slope
The coefficient \(-0.02396\) represents the slope of the line in the linearized form. In this context:
- A negative slope indicates that as \( x \) increases, \( \log y \) decreases. This suggests that \( y \) decreases exponentially with increasing \( x \) when transformed back from logarithmic scale to the original scale.
- Specifically, a slope of \(-0.02396\) means that for each unit increase in \( x \), \( \log y \) decreases by \( 0.02396 \). This implies that \( y \) decreases multiplicatively, corresponding to an exponential decay pattern in the original data set.
### Finding the Value of \( y \) when \( x = 0 \)
To find the value of \( y \) when \( x = 0\), we can substitute \( x = 0 \) into the equation:
\[
\log y = -0.02396(0) + 4.8153
\]
\[
\log y = 4.8153
\]
Now, to convert back from logarithmic form to the original value of \( y \), we can use the property of logarithms:
\[
y = 10^{\log y} = 10^{4.8153}
\]
Now, calculate \( 10^{4.8153} \):
Using a calculator:
\[
y \approx 65557.8
\]
Thus, when \( x = 0 \), the value of \( y \) is approximately \( 65557.8 \).
\[
\log y = -0.02396 x + 4.8153
\]
### Interpretation of the Slope
The coefficient \(-0.02396\) represents the slope of the line in the linearized form. In this context:
- A negative slope indicates that as \( x \) increases, \( \log y \) decreases. This suggests that \( y \) decreases exponentially with increasing \( x \) when transformed back from logarithmic scale to the original scale.
- Specifically, a slope of \(-0.02396\) means that for each unit increase in \( x \), \( \log y \) decreases by \( 0.02396 \). This implies that \( y \) decreases multiplicatively, corresponding to an exponential decay pattern in the original data set.
### Finding the Value of \( y \) when \( x = 0 \)
To find the value of \( y \) when \( x = 0\), we can substitute \( x = 0 \) into the equation:
\[
\log y = -0.02396(0) + 4.8153
\]
\[
\log y = 4.8153
\]
Now, to convert back from logarithmic form to the original value of \( y \), we can use the property of logarithms:
\[
y = 10^{\log y} = 10^{4.8153}
\]
Now, calculate \( 10^{4.8153} \):
Using a calculator:
\[
y \approx 65557.8
\]
Thus, when \( x = 0 \), the value of \( y \) is approximately \( 65557.8 \).