a. Make a scatter plot of the data in the table below.

Question

b. Does it appear that a linear model or a logarithmic model is the better fit for the​ data?
                                                                                
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Question content area bottom
Part 1
a. Choose the scatter plot below.
A.
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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).
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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).
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Answer 1

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.