Question

use the following data set to answer the questions below.
what is the correlation coefficient of this data?
unfortunately, this data set contains an outlier. make a scatterplot of the data to identify the outlier, and then calculate the new correlation coefficient for the data, not including this outlier

Explanation:

Step1: Calculate the correlation coefficient \( r \) for the original data

We can use a statistical software or a calculator with a correlation coefficient function. Inputting the \( x \) and \( y \) values:
\( x = [191.7,48.1,22.1,33.1,37.1,26.7,56.3,64.5,49] \)
\( y=[ - 11257.8,1828.6,1594.9,-1443,694.2,-159.8,1122.9,396.8,-310.7] \)
Using the formula \( r=\frac{n\sum xy-\sum x\sum y}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}} \) (where \( n = 9 \)) or a calculator/software, we find \( r\approx - 0.27 \)

Step2: Identify the outlier

By looking at the scatter - plot (or by comparing the values), the outlier is the point \((191.7,-11257.8)\)

Step3: Calculate the correlation coefficient \( r_{wo} \) without the outlier

Remove the outlier \((191.7,-11257.8)\)
\( x_{new}=[48.1,22.1,33.1,37.1,26.7,56.3,64.5,49] \)
\( y_{new}=[1828.6,1594.9,-1443,694.2,-159.8,1122.9,396.8,-310.7] \)
Using the formula \( r=\frac{n\sum xy-\sum x\sum y}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}} \) (where \( n = 8 \)) or a calculator/software, we find \( r_{wo}\approx0.79 \)

Answer:

The correlation coefficient of the original data \( r\approx - 0.27 \), and the correlation coefficient without the outlier \( r_{wo}\approx0.79 \)