Question

ticket prices for seattle mariners games during the 2008 season. the article included a data set that listed the average ticket price for each mlb team, the league in which the team plays (al or nl), the number of wins during the 2007 season and the cost per win (in dollars). the data for the 16 national league teams are shown below.
\team\tleague\tprice\twins\tcost/win
arizona diamondbacks\tnl\t19.68\t90\t35.40
atlanta braves\tnl\t17.07\t84\t32.89
chicago cubs\tnl\t34.30\t85\t65.33
cincinnati reds\tnl\t17.90\t72\t40.32
colorado rockies\tnl\t14.72\t90\t26.67
florida marlins\tnl\t16.70\t71\t38.13
houston astros\tnl\t26.66\t73\t59.11
los angeles dodgers\tnl\t20.09\t82\t34.64
milwaukee brewers\tnl\t18.11\t83\t35.37
n.y. mets\tnl\t25.28\t88\t46.56
philadelphia phillies\tnl\t26.73\t89\t48.69
pittsburgh pirates\tnl\t17.08\t68\t40.67
san diego padres\tnl\t20.83\t89\t38.15
san francisco giants\tnl\t24.53\t71\t56.00
st. louis cardinals\tnl\t29.78\t78\t61.91
washington nationals\tnl\t20.88\t73\t46.30

compute the correlation between average 2007 price and cost per win for these 16 teams. round your answer to three decimal places.

Explanation:

Step1: Calculate the mean of price ($\bar{x}$) and cost/win ($\bar{y}$)

Let \(x\) be the price and \(y\) be the cost/win.
\(\bar{x}=\frac{19.68 + 17.07+34.30+17.90+14.72+16.70+26.66+20.09+18.11+25.28+26.73+17.08+28.83+24.53+29.78+20.88}{16}\)

$$
LATEXBLOCK0
$$

\(\bar{y}=\frac{35.40+32.89+65.33+40.32+26.67+38.13+59.11+34.64+35.37+46.56+48.69+40.67+38.15+56.00+61.91+46.30}{16}\)

$$
LATEXBLOCK1
$$

Step2: Calculate the numerator of the correlation formula \(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})\)

$$
LATEXBLOCK2
$$

Step3: Calculate the denominator of the correlation formula \(\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}\)

First, \(\sum_{i = 1}^{16}(x_{i}-\bar{x})^{2}=(19.68 - 24.9)^{2}+(17.07 - 24.9)^{2}+\cdots+(20.88 - 24.9)^{2}\)

$$
LATEXBLOCK3
$$

Second, \(\sum_{i = 1}^{16}(y_{i}-\bar{y})^{2}=(35.40 - 42.94625)^{2}+(32.89 - 42.94625)^{2}+\cdots+(46.30 - 42.94625)^{2}\)

$$
LATEXBLOCK4
$$

\(\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}=\sqrt{489.96\times2703.77}\approx\sqrt{1325337.15}\approx1151.23\)

Step4: Calculate the correlation coefficient \(r\)

\(r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}=\frac{1079.93}{1151.23}\approx0.938\)

Answer:

\(0.938\)