Question
a. Make a scatter plot of the data in the table below.
b. Does it appear that a linear model or a logarithmic model is the better fit for the data?
x
1
3
5
7
9
y
7
12
14
15
16
Question content area bottom
Part 1
a. Choose the scatter plot below.
A.
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x
y
A scatterplot has a horizontal x-axis labeled from 0 to 9 in increments of 1 and a vertical y-axis labeled from 0 to 20 in increments of 2. Points are plotted at (1, 7), (3, 12), (5, 14), (7, 15), and (9, 16).
B.
0
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0
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x
y
A scatterplot has a horizontal x-axis labeled from 0 to 20 in increments of 2 and a vertical y-axis labeled from 0 to 9 in increments of 1. Points are plotted at (7, 1), (12, 3), (14, 5), (15, 7), and (16, 9).
C.
0
2
4
6
8
0
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8
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x
y
b. Does it appear that a linear model or a logarithmic model is the better fit for the data?
x
1
3
5
7
9
y
7
12
14
15
16
Question content area bottom
Part 1
a. Choose the scatter plot below.
A.
0
2
4
6
8
0
4
8
12
16
20
x
y
A scatterplot has a horizontal x-axis labeled from 0 to 9 in increments of 1 and a vertical y-axis labeled from 0 to 20 in increments of 2. Points are plotted at (1, 7), (3, 12), (5, 14), (7, 15), and (9, 16).
B.
0
4
8
12
16
20
0
2
4
6
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x
y
A scatterplot has a horizontal x-axis labeled from 0 to 20 in increments of 2 and a vertical y-axis labeled from 0 to 9 in increments of 1. Points are plotted at (7, 1), (12, 3), (14, 5), (15, 7), and (16, 9).
C.
0
2
4
6
8
0
4
8
12
16
20
x
y
Answers
GPT-4o mini
To answer your questions, we'll break it down into parts.
### Part a: Scatter Plot
From the data provided:
- \( x \) values: \( 1, 3, 5, 7, 9 \)
- \( y \) values: \( 7, 12, 14, 15, 16 \)
You can plot these points as follows:
- (1, 7)
- (3, 12)
- (5, 14)
- (7, 15)
- (9, 16)
Based on the descriptions you provided for the scatter plots:
**Choice A** fits the data correctly, as it shows the points plotted at (1, 7), (3, 12), (5, 14), (7, 15), and (9, 16).
### Part b: Model Fit
To evaluate whether a linear model or a logarithmic model is a better fit for the data, we can consider a few observations:
1. **Linear Model**: Linear relationships imply that as \( x \) increases, \( y \) increases at a constant rate. In your data:
- As \( x \) increases from 1 to 9, \( y \) increases as follows:
- From \( 7 \) (at \( x = 1 \)) to \( 12 \) (at \( x = 3 \)): Increase of 5
- From \( 12 \) to \( 14 \) (at \( x = 5 \)): Increase of 2
- From \( 14 \) to \( 15 \) (at \( x = 7 \)): Increase of 1
- From \( 15 \) to \( 16 \) (at \( x = 9 \)): Increase of 1
The increments are declining, which suggests that the relationship is not strictly linear.
2. **Logarithmic Model**: A logarithmic model might provide a better fit when the rate of change decreases as \( x \) increases. Given that the increments in \( y \) are decreasing (from 5 to 1), a logarithmic model might fit the data more closely.
### Conclusion:
To conclude, just by visual inspection:
- **Linear Fit**: The points appear somewhat linear but deviate as \( x \) increases.
- **Logarithmic Fit**: May capture the trends better as the rate of change in \( y \) decreases.
Therefore, a **logarithmic model** may be the better fit for the data as it better accommodates the declining rate of increase in \( y \) with increasing \( x \).
For a definitive conclusion, you would typically perform regression analysis to assess the goodness-of-fit for each model quantitatively.
### Part a: Scatter Plot
From the data provided:
- \( x \) values: \( 1, 3, 5, 7, 9 \)
- \( y \) values: \( 7, 12, 14, 15, 16 \)
You can plot these points as follows:
- (1, 7)
- (3, 12)
- (5, 14)
- (7, 15)
- (9, 16)
Based on the descriptions you provided for the scatter plots:
**Choice A** fits the data correctly, as it shows the points plotted at (1, 7), (3, 12), (5, 14), (7, 15), and (9, 16).
### Part b: Model Fit
To evaluate whether a linear model or a logarithmic model is a better fit for the data, we can consider a few observations:
1. **Linear Model**: Linear relationships imply that as \( x \) increases, \( y \) increases at a constant rate. In your data:
- As \( x \) increases from 1 to 9, \( y \) increases as follows:
- From \( 7 \) (at \( x = 1 \)) to \( 12 \) (at \( x = 3 \)): Increase of 5
- From \( 12 \) to \( 14 \) (at \( x = 5 \)): Increase of 2
- From \( 14 \) to \( 15 \) (at \( x = 7 \)): Increase of 1
- From \( 15 \) to \( 16 \) (at \( x = 9 \)): Increase of 1
The increments are declining, which suggests that the relationship is not strictly linear.
2. **Logarithmic Model**: A logarithmic model might provide a better fit when the rate of change decreases as \( x \) increases. Given that the increments in \( y \) are decreasing (from 5 to 1), a logarithmic model might fit the data more closely.
### Conclusion:
To conclude, just by visual inspection:
- **Linear Fit**: The points appear somewhat linear but deviate as \( x \) increases.
- **Logarithmic Fit**: May capture the trends better as the rate of change in \( y \) decreases.
Therefore, a **logarithmic model** may be the better fit for the data as it better accommodates the declining rate of increase in \( y \) with increasing \( x \).
For a definitive conclusion, you would typically perform regression analysis to assess the goodness-of-fit for each model quantitatively.