find the correlation coefficient, r, of the data described below.\na television sports commentator recently…

find the correlation coefficient, r, of the data described below.\na television sports commentator recently declared that basketball teams that rely on the three - point shot cannot win championships. to see if the commentator had a valid point, a sports analyst reviewed game statistics from several teams over the past season.\nfor each of the teams, the analyst calculated what percentage of attempted field goals had been three - pointers, x, and the number of wins in the season, y.\npercentage of shots taken from three - point range|number of wins in a season\n20|37\n23|21\n26|60\n28|35\n29|30

find the correlation coefficient, r, of the data described below.\na television sports commentator recently declared that basketball teams that rely on the three - point shot cannot win championships. to see if the commentator had a valid point, a sports analyst reviewed game statistics from several teams over the past season.\nfor each of the teams, the analyst calculated what percentage of attempted field goals had been three - pointers, x, and the number of wins in the season, y.\npercentage of shots taken from three - point range|number of wins in a season\n20|37\n23|21\n26|60\n28|35\n29|30

Answer

Explanation:

Step1: Calculate the means

Let $x = [20,23,26,28,29]$ and $y=[37,21,60,35,30]$. The mean of $x$, $\bar{x}=\frac{20 + 23+26+28+29}{5}=\frac{126}{5}=25.2$. The mean of $y$, $\bar{y}=\frac{37+21+60+35+30}{5}=\frac{183}{5}=36.6$.

Step2: Calculate the numerator and denominators of the correlation - coefficient formula

The formula for the correlation coefficient $r=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_i-\bar{x})^2\sum_{i = 1}^{n}(y_i - \bar{y})^2}}$. Calculate $(x_i-\bar{x})(y_i - \bar{y})$ for each $i$: For $i = 1$: $(20 - 25.2)(37-36.6)=-5.2\times0.4=-2.08$. For $i = 2$: $(23 - 25.2)(21 - 36.6)=-2.2\times(-15.6) = 34.32$. For $i = 3$: $(26 - 25.2)(60 - 36.6)=0.8\times23.4 = 18.72$. For $i = 4$: $(28 - 25.2)(35 - 36.6)=2.8\times(-1.6)=-4.48$. For $i = 5$: $(29 - 25.2)(30 - 36.6)=3.8\times(-6.6)=-25.08$. $\sum_{i = 1}^{5}(x_i-\bar{x})(y_i - \bar{y})=-2.08 + 34.32+18.72-4.48-25.08=1.4$. Calculate $(x_i-\bar{x})^2$ for each $i$: $(20 - 25.2)^2=(-5.2)^2 = 27.04$. $(23 - 25.2)^2=(-2.2)^2 = 4.84$. $(26 - 25.2)^2=(0.8)^2 = 0.64$. $(28 - 25.2)^2=(2.8)^2 = 7.84$. $(29 - 25.2)^2=(3.8)^2 = 14.44$. $\sum_{i = 1}^{5}(x_i-\bar{x})^2=27.04 + 4.84+0.64+7.84+14.44=54.8$. Calculate $(y_i - \bar{y})^2$ for each $i$: $(37 - 36.6)^2=(0.4)^2 = 0.16$. $(21 - 36.6)^2=(-15.6)^2 = 243.36$. $(60 - 36.6)^2=(23.4)^2 = 547.56$. $(35 - 36.6)^2=(-1.6)^2 = 2.56$. $(30 - 36.6)^2=(-6.6)^2 = 43.56$. $\sum_{i = 1}^{5}(y_i - \bar{y})^2=0.16+243.36+547.56+2.56+43.56=837.2$. The denominator is $\sqrt{54.8\times837.2}=\sqrt{45878.56}\approx214.19$.

Step3: Calculate the correlation coefficient

$r=\frac{1.4}{214.19}\approx0.0065$.

Answer:

$r\approx0.0065$