Question

for the accompanying data set, (a) draw a scatter diagram of the data, (b) compute the correlation coefficient, and (c) determine whether there is a linear relation between ( x ) and ( y ).
click here to view the data set.
click here to view the critical values table.
(a) draw a scatter diagram of the data. choose the correct graph.
(b) compute the correlation coefficient.
the correlation coefficient is ( r = 0.396 ). (round to three decimal places as needed.)
(c) determine whether there is a linear relation between ( x ) and ( y ).
because the correlation coefficient is (square) and the absolute value of the correlation coefficient, (square), is (square) than the critical value for this data set, (square) linear relation exists between ( x ) and ( y ).
(round to three decimal places as needed.)

Explanation:

Step1: Calculate the means

For \(x\): \(\bar{x}=\frac{2 + 6+1+7+9}{5}=\frac{25}{5} = 5\)
For \(y\): \(\bar{y}=\frac{3 + 7+6+9+5}{5}=\frac{30}{5}=6\)

Step2: Calculate numerator and denominator

Numerator: \(\sum_{i = 1}^{5}(x_{i}-\bar{x})(y_{i}-\bar{y})=(2 - 5)(3 - 6)+(6 - 5)(7 - 6)+(1 - 5)(6 - 6)+(7 - 5)(9 - 6)+(9 - 5)(5 - 6)\)
\(=(- 3)\times(-3)+1\times1+(-4)\times0+2\times3+4\times(-1)\)
\(=9 + 1+0 + 6-4=12\)
Denominator: \(\sqrt{\sum_{i = 1}^{5}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{5}(y_{i}-\bar{y})^{2}}\)
\(\sum_{i = 1}^{5}(x_{i}-\bar{x})^{2}=(2 - 5)^{2}+(6 - 5)^{2}+(1 - 5)^{2}+(7 - 5)^{2}+(9 - 5)^{2}\)
\(=9+1 + 16+4+16=46\)
\(\sum_{i = 1}^{5}(y_{i}-\bar{y})^{2}=(3 - 6)^{2}+(7 - 6)^{2}+(6 - 6)^{2}+(9 - 6)^{2}+(5 - 6)^{2}\)
\(=9+1+0 + 9+1=20\)
\(\sqrt{46\times20}=\sqrt{920}\approx30.33\)

Step3: Calculate correlation coefficient

\(r=\frac{\sum_{i = 1}^{5}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{5}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{5}(y_{i}-\bar{y})^{2}}}=\frac{12}{30.33}\approx0.396\)
Assume the critical value (for \(n = 5\)) from the critical - values table is \(0.878\) (common critical value for a significance level of \(0.05\) in a two - tailed test for simple linear regression with \(n=5\)).
Since \(|r|=|0.396|\) and \(0.396<0.878\)

Answer:

(b) The correlation coefficient \(r = 0.396\)
(c) Because the correlation coefficient is positive and the absolute value of the correlation coefficient, \(0.396\), is less than the critical value for this data set, \(0.878\), no linear relation exists between \(x\) and \(y\).