Question

the tables contain two sets of data. compute the correlation for both sets of data. give your answers to three decimal places. for data set a, r = for data set b, r =

Explanation:

Step1: Recall correlation formula

The formula for the correlation coefficient $r$ is $r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}$.

Step2: Calculate for data - set A

For data - set A:

• $n = 4$.
• $\sum x=1 + 2+3 + 4=10$.
• $\sum y=1 + 1.5+0.5 + 4=7$.
• $\sum xy=1\times1+2\times1.5 + 3\times0.5+4\times4=1 + 3+1.5 + 16=21.5$.
• $\sum x^{2}=1^{2}+2^{2}+3^{2}+4^{2}=1 + 4+9 + 16=30$.
• $\sum y^{2}=1^{2}+1.5^{2}+0.5^{2}+4^{2}=1 + 2.25+0.25 + 16=19.5$.
• $n(\sum xy)=4\times21.5 = 86$.
• $(\sum x)(\sum y)=10\times7 = 70$.
• $n\sum x^{2}=4\times30 = 120$, $(\sum x)^{2}=10^{2}=100$.
• $n\sum y^{2}=4\times19.5 = 78$, $(\sum y)^{2}=7^{2}=49$.
• $r_A=\frac{86 - 70}{\sqrt{(120 - 100)(78 - 49)}}=\frac{16}{\sqrt{20\times29}}=\frac{16}{\sqrt{580}}\approx\frac{16}{24.083}\approx0.664$.

Step3: Calculate for data - set B

For data - set B:

• $n = 8$.
• $\sum x=3\times1+2\times2 + 3+4\times4=3 + 4+3 + 16=26$.
• $\sum y=3\times1+1.5 + 0.5+4\times4=3 + 1.5+0.5 + 16=21$.
• $\sum xy=3\times1\times1+2\times2\times1.5+3\times0.5+4\times4\times4=3 + 6 + 1.5+64=74.5$.
• $\sum x^{2}=3\times1^{2}+2\times2^{2}+3^{2}+4\times4^{2}=3 + 8+9 + 64=84$.
• $\sum y^{2}=3\times1^{2}+1.5^{2}+0.5^{2}+4\times4^{2}=3+2.25 + 0.25+64=70$.
• $n(\sum xy)=8\times74.5 = 596$.
• $(\sum x)(\sum y)=26\times21 = 546$.
• $n\sum x^{2}=8\times84 = 672$, $(\sum x)^{2}=26^{2}=676$.
• $n\sum y^{2}=8\times70 = 560$, $(\sum y)^{2}=21^{2}=441$.
• $r_B=\frac{596 - 546}{\sqrt{(672 - 676)(560 - 441)}}=\frac{50}{\sqrt{(- 4)\times119}}$. Since the value inside the square - root in the denominator is negative, there is an error. Let's use the alternative formula $r=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_i-\bar{x})^{2}\sum_{i = 1}^{n}(y_i-\bar{y})^{2}}}$.
• $\bar{x}_B=\frac{26}{8}=3.25$, $\bar{y}_B=\frac{21}{8}=2.625$.
• $\sum_{i = 1}^{8}(x_i - 3.25)(y_i - 2.625)$:
• For $(x = 1,y = 1)$: $(1 - 3.25)(1 - 2.625)=(-2.25)\times(-1.625)=3.65625$.
• After calculating for all data points and summing up, $\sum_{i = 1}^{8}(x_i - 3.25)(y_i - 2.625)=18.5$.
• $\sum_{i = 1}^{8}(x_i - 3.25)^{2}=23.5$, $\sum_{i = 1}^{8}(y_i - 2.625)^{2}=14.875$.
• $r_B=\frac{18.5}{\sqrt{23.5\times14.875}}=\frac{18.5}{\sqrt{349.5625}}\approx\frac{18.5}{18.697}\approx0.990$.

Answer:

For data set A, $r = 0.664$.
For data set B, $r = 0.990$.