silver springs moving and storage incorporated is studying the relationship between the number of rooms in a…
silver springs moving and storage incorporated is studying the relationship between the number of rooms in a move and the number of labor hours required for the move. compute a correlation coefficient.\nrooms labor hours rooms labor hours\n1.0 3 2.5 17\n1.0 15 3.0 18\n1.5 8 3.0 35\n1.5 16 3.5 28\n2.0 17 4.0 19\n2.0 15 4.5 33\n2.5 16 5.0 40\n2.5 24\nnote: round your answer to 2 decimal places.\ncorrelation coefficient
Answer
Explanation:
Step1: Calculate the means
Let $x$ be the number of rooms and $y$ be the labor - hours. First, find the mean of $x$, $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$ and the mean of $y$, $\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}$. $n = 14$. $\sum_{i=1}^{14}x_{i}=1.0 + 1.0+1.5+1.5+2.0+2.0+2.5+2.5+2.5+3.0+3.0+3.5+4.0+4.5+5.0=36$ $\bar{x}=\frac{36}{14}\approx2.57$ $\sum_{i = 1}^{14}y_{i}=3 + 15+8+16+17+15+16+24+17+18+35+28+19+33+40 = 284$ $\bar{y}=\frac{284}{14}\approx20.29$
Step2: Calculate the numerator and denominator
The correlation coefficient $r$ is given by the formula: $r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}$ Calculate $(x_{i}-\bar{x})(y_{i}-\bar{y})$, $(x_{i}-\bar{x})^{2}$ and $(y_{i}-\bar{y})^{2}$ for each $i$: For example, when $x_1 = 1.0$ and $y_1 = 3$: $(x_1-\bar{x})=(1.0 - 2.57)=- 1.57$, $(y_1-\bar{y})=(3 - 20.29)=-17.29$ $(x_1-\bar{x})(y_1-\bar{y})=(-1.57)\times(-17.29)=27.1453$ $(x_1 - \bar{x})^{2}=(-1.57)^{2}=2.4649$ $(y_1-\bar{y})^{2}=(-17.29)^{2}=299.9441$ After calculating for all $i$ from $1$ to $14$: $\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})=177.57$ $\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=13.71$ $\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}=2376.86$ $r=\frac{177.57}{\sqrt{13.71\times2376.86}}=\frac{177.57}{\sqrt{32687.75}}=\frac{177.57}{180.79}=0.98$
Answer:
$0.98$