what is the correlation coefficient for the data shown in the table?\n0 0\n1 1\n4 4\n5 5\n0\n1\n4\n5

Explanation:

Step1: Recall correlation coefficient formula

The correlation coefficient $r=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n}(x_i-\bar{x})^2\sum_{i = 1}^{n}(y_i-\bar{y})^2}}$. First, find the means of $x$ and $y$. For $x$ - values $x = {0,1,4,5}$, $\bar{x}=\frac{0 + 1+4+5}{4}=\frac{10}{4}=2.5$. For $y$ - values $y={0,1,4,5}$, $\bar{y}=\frac{0 + 1+4+5}{4}=2.5$.

Step2: Calculate numerator

$(0 - 2.5)(0 - 2.5)+(1 - 2.5)(1 - 2.5)+(4 - 2.5)(4 - 2.5)+(5 - 2.5)(5 - 2.5)$ $=(- 2.5)\times(-2.5)+(-1.5)\times(-1.5)+1.5\times1.5 + 2.5\times2.5$ $=6.25+2.25 + 2.25+6.25=17$.

Step3: Calculate denominator - part 1

$(0 - 2.5)^2+(1 - 2.5)^2+(4 - 2.5)^2+(5 - 2.5)^2$ $=(-2.5)^2+(-1.5)^2+1.5^2+2.5^2$ $=6.25 + 2.25+2.25+6.25 = 17$.

Step4: Calculate denominator - part 2

It is the same as part 1 since $x$ and $y$ values are identical, so $\sum_{i = 1}^{n}(y_i-\bar{y})^2 = 17$. The denominator $\sqrt{17\times17}=17$.

Step5: Calculate correlation coefficient

$r=\frac{17}{17}=1$.

Answer:

B. 1