Question

1. * required the table shows a set of data. the data set is modeled with the linear best - fit model. what is the correlation coefficient? (round your answer to 3 decimal places.) type a response x: 90, 87, 83, 92, 85 y: 80, 89, 94, 82, 91

Explanation:

Step1: List the data points

We have the following pairs of \((x, y)\) values: \((90, 80)\), \((87, 89)\), \((83, 94)\), \((92, 82)\), \((85, 91)\)

Step2: Calculate the necessary sums

First, we calculate the means of \(x\) and \(y\).
Let \(x\) values be \(x_1 = 90\), \(x_2 = 87\), \(x_3 = 83\), \(x_4 = 92\), \(x_5 = 85\)
Let \(y\) values be \(y_1 = 80\), \(y_2 = 89\), \(y_3 = 94\), \(y_4 = 82\), \(y_5 = 91\)

Mean of \(x\), \(\bar{x}=\frac{\sum_{i = 1}^{5}x_i}{5}=\frac{90 + 87+83 + 92+85}{5}=\frac{437}{5}=87.4\)

Mean of \(y\), \(\bar{y}=\frac{\sum_{i = 1}^{5}y_i}{5}=\frac{80 + 89+94 + 82+91}{5}=\frac{436}{5}=87.2\)

Now, we calculate \(\sum_{i = 1}^{5}(x_i-\bar{x})(y_i - \bar{y})\), \(\sum_{i = 1}^{5}(x_i-\bar{x})^2\) and \(\sum_{i = 1}^{5}(y_i-\bar{y})^2\)

For \((x_1-\bar{x})(y_1 - \bar{y})=(90 - 87.4)(80 - 87.2)=(2.6)(- 7.2)=-18.72\)

\((x_2-\bar{x})(y_2 - \bar{y})=(87 - 87.4)(89 - 87.2)=(-0.4)(1.8)=-0.72\)

\((x_3-\bar{x})(y_3 - \bar{y})=(83 - 87.4)(94 - 87.2)=(-4.4)(6.8)=-29.92\)

\((x_4-\bar{x})(y_4 - \bar{y})=(92 - 87.4)(82 - 87.2)=(4.6)(-5.2)=-23.92\)

\((x_5-\bar{x})(y_5 - \bar{y})=(85 - 87.4)(91 - 87.2)=(-2.4)(3.8)=-9.12\)

\(\sum_{i = 1}^{5}(x_i-\bar{x})(y_i - \bar{y})=-18.72-0.72 - 29.92-23.92-9.12=-82.4\)

For \((x_i-\bar{x})^2\):

\((90 - 87.4)^2=(2.6)^2 = 6.76\)

\((87 - 87.4)^2=(-0.4)^2=0.16\)

\((83 - 87.4)^2=(-4.4)^2 = 19.36\)

\((92 - 87.4)^2=(4.6)^2=21.16\)

\((85 - 87.4)^2=(-2.4)^2 = 5.76\)

\(\sum_{i = 1}^{5}(x_i-\bar{x})^2=6.76+0.16+19.36 + 21.16+5.76=53.2\)

For \((y_i-\bar{y})^2\):

\((80 - 87.2)^2=(-7.2)^2=51.84\)

\((89 - 87.2)^2=(1.8)^2 = 3.24\)

\((94 - 87.2)^2=(6.8)^2=46.24\)

\((82 - 87.2)^2=(-5.2)^2=27.04\)

\((91 - 87.2)^2=(3.8)^2 = 14.44\)

\(\sum_{i = 1}^{5}(y_i-\bar{y})^2=51.84+3.24+46.24+27.04 + 14.44=142.8\)

Step3: Calculate the correlation coefficient \(r\)

The formula for the correlation coefficient is \(r=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_i-\bar{x})^2\sum_{i = 1}^{n}(y_i-\bar{y})^2}}\)

Substitute the values:

\(r=\frac{- 82.4}{\sqrt{53.2\times142.8}}\)

First, calculate \(53.2\times142.8 = 53.2\times142.8 = 7596.96\)

Then, \(\sqrt{7596.96}\approx87.16\)

Then, \(r=\frac{-82.4}{87.16}\approx - 0.945\)

Answer:

\(-0.945\)