Make a scatterplot of the data with two new points added. a) 10% return, 25 new birds. b) 40% return, 5 new birds. Find two new correlations: for the original data plus Point A and for the original data plus Point B.

Make a scatterplot of the data with two new points added. a) 10% return, 25 new birds. b) 40% return, 5 new birds. Find two new correlations: for the original data plus Point A and for the original data plus Point B.

Question
Scatterplots
asked 2020-12-07
Make a scatterplot of the data with two new points added.
a) \(\displaystyle{10}\%\) return, 25 new birds.
b) \(\displaystyle{40}\%\) return, 5 new birds.
Find two new correlations: for the original data plus Point A and for the original data plus Point B.

Answers (1)

2020-12-08

Step 1
The recentage return is on the horizontal axis and the new birds are on the vertical axis.
image
Step 2
Correlation coefficient for the original data including point A.
Find \(\displaystyle{X}\ \cdot\ {Y},\ {X}^{{{2}}}\ {\quad\text{and}\quad}\ {Y}^{{{2}}}\) as it was done in the table below.
\(\begin{array}{|c|c|}\hline X & Y & X\ \cdot\ Y & X\ \cdot\ X & Y\ \cdot\ Y\ \\ \hline 74.0000000000000 & 5.00000000000000 & 370 & 5476 & 25 \\ \hline 66.0000000000000 & 6.00000000000000 & 396 & 4356 & 36 \\ \hline 81.0000000000000 & 8.00000000000000 & 648 & 6561 & 64 \\ \hline 52.0000000000000 & 11.0000000000000 & 572 & 2704 & 121 \\ \hline 73.0000000000000 & 12.0000000000000 & 876 & 5329 & 144 \\ \hline 62.0000000000000 & 15.0000000000000 & 930 & 3844 & 225 \\ \hline 52.0000000000000 & 16.0000000000000 & 832 & 2704 & 256 \\ \hline 45.0000000000000 & 17.0000000000000 & 765 & 2025 & 289 \\ \hline 62.0000000000000 & 18.0000000000000 & 1116 & 3844 & 324 \\ \hline 46.0000000000000 & 18.0000000000000 & 828 & 2116 & 324 \\ \hline 60.0000000000000 & 19.0000000000000 & 1140 & 3600 & 361 \\ \hline 46.0000000000000 & 20.0000000000000 & 920 & 2116 & 400 \\ \hline 38.0000000000000 & 20.0000000000000 & 760 & 1444 & 400 \\ \hline 10.0000000000000 & 25.0000000000000 & 250 & 100 & 625 \\ \hline \end{array}\)
Find the sum of every column to get:
\(\displaystyle\sum\ {X}={767},\ \sum\ {Y}={210},\ \sum\ {X}\ \cdot\ {Y}={10403},\ \sum\ {X}^{{{2}}}={46219},\ \sum\ {Y}^{{{2}}}={3594}\)
Use the following formula to work out the correlation coefficient.
\(\displaystyle{r}={\frac{{{n}\ \cdot\ \sum\ {X}{Y}\ -\ \sum\ {X}\ \cdot\ \sum\ {Y}}}{{\sqrt{{{\left[{n}\ \sum\ {X}^{{{2}}}\ -\ {\left(\sum\ {X}\right)}^{{{2}}}\right]}\ \cdot\ {\left[{n}\ \sum\ {Y}^{{{2}}}\ -\ {\left(\sum\ {Y}\right)}^{{{2}}}\right]}}}}}}\)
\(\displaystyle{r}={\frac{{{14}\ \cdot\ {10403}\ -\ {767}\ \cdot\ {210}}}{{\sqrt{{{\left[{14}\ \cdot\ {46219}\ -\ {767}^{{{2}}}\right]}\ \cdot\ {\left[{14}\ \cdot\ {3594}\ -\ {210}^{{{2}}}\right]}}}}}}\ \approx\ -{0.8071}\)
Step 3
Correlation coefficient for the data including point B.
Find \(\displaystyle{X}\ \cdot\ {Y},\ {X}^{{{2}}}\ {\quad\text{and}\quad}\ {Y}^{{{2}}}\) as it was done in the table below.
\(\begin{array}{|c|c|}\hline X & Y & X\ \cdot\ Y & X\ \cdot\ X & Y\ \cdot\ Y\ \\ \hline 74.0000000000000 & 5.00000000000000 & 370 & 5476 & 25 \\ \hline 66.0000000000000 & 6.00000000000000 & 396 & 4356 & 36 \\ \hline 81.0000000000000 & 8.00000000000000 & 648 & 6561 & 64 \\ \hline 52.0000000000000 & 11.0000000000000 & 572 & 2704 & 121 \\ \hline 73.0000000000000 & 12.0000000000000 & 876 & 5329 & 144 \\ \hline 62.0000000000000 & 15.0000000000000 & 930 & 3844 & 225 \\ \hline 52.0000000000000 & 16.0000000000000 & 832 & 2704 & 256 \\ \hline 45.0000000000000 & 17.0000000000000 & 765 & 2025 & 289 \\ \hline 62.0000000000000 & 18.0000000000000 & 1116 & 3844 & 324 \\ \hline 46.0000000000000 & 18.0000000000000 & 828 & 2116 & 324 \\ \hline 60.0000000000000 & 19.0000000000000 & 1140 & 3600 & 361 \\ \hline 46.0000000000000 & 20.0000000000000 & 920 & 2116 & 400 \\ \hline 38.0000000000000 & 20.0000000000000 & 760 & 1444 & 400 \\ \hline 40.0000000000000 & 5.00000000000000 & 200 & 1600 & 25 \\ \hline \end{array}\)
Find the sum of every column to get:
\(\displaystyle\sum\ {X}={797},\ \sum\ {Y}={190},\ \sum\ {X}\ \cdot\ {Y}={10353},\ \sum\ {X}^{{{2}}}={47719},\ \sum\ {Y}^{{{2}}}={2994}\)
Use the following formula to work out the correlation coefficient.
\(\displaystyle{r}={\frac{{{n}\ \cdot\ \sum\ {X}{Y}\ -\ \sum\ {X}\ \cdot\ \sum\ {Y}}}{{\sqrt{{{\left[{n}\ \sum\ {X}^{{{2}}}\ -\ {\left(\sum\ {X}\right)}^{{{2}}}\right]}\ \cdot\ {\left[{n}\ \sum\ {Y}^{{{2}}}\ -\ {\left(\sum\ {Y}\right)}^{{{2}}}\right]}}}}}}\)
\(\displaystyle{r}={\frac{{{14}\ \cdot\ {10353}\ -\ {797}\ \cdot\ {190}}}{{\sqrt{{{\left[{14}\ \cdot\ {47719}\ -\ {797}^{{{2}}}\right]}\ \cdot\ {\left[{14}\ \cdot\ {2994}\ -\ {190}^{{{2}}}\right]}}}}}}\ \approx\ -{0.4693}\)

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