Question

the data in the table to the right are based on the results of a survey comparing the commute time of adults to their score on a well - being test. complete parts (a) through (d). click the icon to view the table of critical values of the correlation coefficient. (d) does a linear relation exist between the commute time and well - being index score? select the correct choice and fill in the answer box to complete your choice. (round to three decimal places as needed.) a. no, the variables commute time and well - being score are not linearly related because r is positive and the absolute value of the correlation coefficient is less than the critical value. b. yes, the variables commute time and well - being score are positively associated because r is negative and the absolute value of the correlation coefficient is greater than the critical value. c. yes, the variables commute time and well - being score are negatively associated because r is negative and the absolute value of the correlation coefficient is greater than the critical value. d. no, the variables commute time and well - being score are not linearly related because r is negative and the absolute value of the correlation coefficient is less than the critical value. e. yes, the variables commute time and well - being score are negatively associated because r is positive and the absolute value of the correlation coefficient is greater than the critical value.

Explanation:

Step1: Calculate the correlation coefficient \( r \)

Using the formula for the correlation coefficient \( r=\frac{n\sum xy - (\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}\)
Let \( x \) be commute time and \( y \) be well - being score.
\( n = 7 \)
\(\sum x=5 + 14+25+34+50+73+105=306\)
\(\sum y=69.8+68.6+67.6+67.1+66.5+66.4+63.5 = 469.5\)
\(\sum xy=(5\times69.8)+(14\times68.6)+(25\times67.6)+(34\times67.1)+(50\times66.5)+(73\times66.4)+(105\times63.5)\)
\(=349+960.4+1690+2281.4+3325+4847.2+6667.5=20120.5\)
\(\sum x^{2}=5^{2}+14^{2}+25^{2}+34^{2}+50^{2}+73^{2}+105^{2}\)
\(=25 + 196+625+1156+2500+5329+11025=20856\)
\(\sum y^{2}=69.8^{2}+68.6^{2}+67.6^{2}+67.1^{2}+66.5^{2}+66.4^{2}+63.5^{2}\)
\(=4872.04+4705.96+4569.76+4502.41+4422.25+4408.96+4032.25=31513.63\)

Substitute into the formula:

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Assume the critical value for \( n = 7\) (degree of freedom \(n - 2=5\)) is \(r_{c}=0.754\) (from the table of critical values of the correlation coefficient)

Since \(|r|=|- 0.960|>r_{c} = 0.754\) and \(r<0\)

Answer:

C. Yes, the variables commute time and well - being score are negatively associated because \(r\) is negative and the absolute value of the correlation coefficient is greater than the critical value.