Question

use the given data to complete parts (a) and (b).

compute the linear correlation coefficient with the additional data point. the linear correlation coefficient for the five pieces of data is 0.883
(round to three decimal places as needed.)
comment on the effect the additional data point has on the linear correlation coefficient.

a. the additional data point strengthens the appearance of a linear association between the data points.
b. the additional data point does not affect the linear correlation coefficient.
c. the additional data point weakens the appearance of a linear association between the data points.

Explanation:

Step1: Recall the property of linear correlation coefficient

The linear correlation coefficient \(r\) measures the strength and direction of a linear relationship between two variables. The value of \(r\) ranges from \(- 1\) to \(1\). A value of \(r\) close to \(1\) or \(-1\) indicates a strong linear relationship, while a value close to \(0\) indicates a weak linear relationship.

Step2: Analyze the change in \(r\)

The original number of data points \(n_1 = 4\). After adding a data - point, \(n_2=5\). The formula for the linear correlation coefficient is \(r=\frac{n\sum xy-\sum x\sum y}{\sqrt{n\sum x^{2}-(\sum x)^{2}}\sqrt{n\sum y^{2}-(\sum y)^{2}}}\).
Let's assume the additional data point is \((x_0,y_0)\). When we calculate \(r\) for \(n = 5\) data points \((2.1,3.8),(3.8,1.5),(3,3.6),(4.7,4.9),(x_0,y_0)\) (we can assume the new data point is \((11,9)\) from the first scatter - plot).
\(\sum x=2.1 + 3.8+3 + 4.7+11=24.6\), \(\sum y=3.8 + 1.5+3.6+4.9+9=22.8\), \(\sum xy=(2.1\times3.8)+(3.8\times1.5)+(3\times3.6)+(4.7\times4.9)+(11\times9)=7.98+5.7 + 10.8+23.03+99=146.51\), \(\sum x^{2}=2.1^{2}+3.8^{2}+3^{2}+4.7^{2}+11^{2}=4.41+14.44+9+22.09+121 = 170.94\), \(\sum y^{2}=3.8^{2}+1.5^{2}+3.6^{2}+4.9^{2}+9^{2}=14.44+2.25+12.96+24.01+81=134.66\)

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Since \(0.877<0.883\) (the correlation coefficient for \(n = 4\) data points), the additional data point weakens the appearance of a linear association between the data points.

Answer:

C. The additional data point weakens the appearance of a linear association between the data points.