calculate the correlation coefficient r, letting row 1 represent the x - values and row 2 the y - values…

calculate the correlation coefficient r, letting row 1 represent the x - values and row 2 the y - values. then calculate it again, letting row 2 represent the x - values and row 1 the y - values. what effect does switching the variables have on r?\nrow 1: 14 25 36 41 56 63 71\nrow 2: 119 187 120 183 157 192 199\ncalculate the correlation coefficient r, letting row 1 represent the x - values and row 2 the y - values\nr = (round to three decimal places as needed)

calculate the correlation coefficient r, letting row 1 represent the x - values and row 2 the y - values. then calculate it again, letting row 2 represent the x - values and row 1 the y - values. what effect does switching the variables have on r?\nrow 1: 14 25 36 41 56 63 71\nrow 2: 119 187 120 183 157 192 199\ncalculate the correlation coefficient r, letting row 1 represent the x - values and row 2 the y - values\nr = (round to three decimal places as needed)

Answer

Explanation:

Step1: Calculate the means of x and y

Let $x = [14,25,36,41,56,63,71]$, $y=[119,187,120,183,157,192,199]$. The mean of $x$, $\bar{x}=\frac{14 + 25+36+41+56+63+71}{7}=\frac{306}{7}\approx43.714$. The mean of $y$, $\bar{y}=\frac{119+187+120+183+157+192+199}{7}=\frac{1157}{7}\approx165.286$.

Step2: Calculate the numerator of the correlation - coefficient formula

[ \begin{align*} \sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})&=(14 - 43.714)(119-165.286)+(25 - 43.714)(187 - 165.286)+(36-43.714)(120 - 165.286)+(41-43.714)(183 - 165.286)+(56 - 43.714)(157-165.286)+(63-43.714)(192 - 165.286)+(71-43.714)(199 - 165.286)\ &=(-29.714)(-46.286)+(-18.714)(21.714)+(-7.714)(-45.286)+(-2.714)(17.714)+(12.286)(-8.286)+(19.286)(26.714)+(27.286)(33.714)\ &=1375.17+(-406.34)+349.47+(-47.07)+(-101.99)+515.29+919.12\ &=2503.65 \end{align*} ]

Step3: Calculate the denominator of the correlation - coefficient formula

[ \begin{align*} \sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\sqrt{\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}&=\sqrt{(14 - 43.714)^{2}+(25 - 43.714)^{2}+(36-43.714)^{2}+(41-43.714)^{2}+(56 - 43.714)^{2}+(63-43.714)^{2}+(71-43.714)^{2}}\times\sqrt{(119 - 165.286)^{2}+(187 - 165.286)^{2}+(120 - 165.286)^{2}+(183 - 165.286)^{2}+(157-165.286)^{2}+(192 - 165.286)^{2}+(199 - 165.286)^{2}}\ &=\sqrt{(-29.714)^{2}+(-18.714)^{2}+(-7.714)^{2}+(-2.714)^{2}+(12.286)^{2}+(19.286)^{2}+(27.286)^{2}}\times\sqrt{(-46.286)^{2}+(21.714)^{2}+(-45.286)^{2}+(17.714)^{2}+(-8.286)^{2}+(26.714)^{2}+(33.714)^{2}}\ &=\sqrt{882.93 + 349.21+59.51+7.37+150.97+372.97+744.57}\times\sqrt{2142.47+471.41+2050.87+313.71+68.66+713.67+1136.67}\ &=\sqrt{2566.53}\times\sqrt{6897.46}\ &=50.661\times83.051\ & = 4207.77 \end{align*} ]

Step4: Calculate the correlation coefficient r

[r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\sqrt{\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}=\frac{2503.65}{4207.77}\approx0.595]

When we switch the variables (letting Row 2 be $x$ - values and Row 1 be $y$ - values), the correlation coefficient $r$ remains the same. This is because the correlation coefficient measures the strength and direction of the linear relationship between two variables, and the formula for $r$ is symmetric with respect to $x$ and $y$.

Answer:

$0.595$