The data below are the temperatures on randomly chosen days during a summer class and the number of absences on those days. Calculate the correlation coefficient, r. $$\begin{array} { l | c | c | c | c | c | c | c | c | c | }
\text { Temperature, } \mathrm { x } & 76 & 89 & 95 & 94 & 92 & 102 & 79 & 104 & 84 \
\hline \text { Number of absences, } y & 12 & 16 & 19 & 19 & 17 & 24 & 13 & 24 & 14
\end{array}$$
A)0.980
B)0.890
C)0.881
D)0.819

Question

The data below are the temperatures on randomly chosen days during a summer class and the number of absences on those days. Calculate the correlation coefficient, r. $$\begin{array} { l | c | c | c | c | c | c | c | c | c | } 
\text { Temperature, } \mathrm { x } & 76 & 89 & 95 & 94 & 92 & 102 & 79 & 104 & 84 \
\hline \text { Number of absences, } y & 12 & 16 & 19 & 19 & 17 & 24 & 13 & 24 & 14
\end{array}$$
A)0.980
B)0.890
C)0.881
D)0.819

Quizplus · Accepted Answer

The correlation coefficient, r, is a measure of the strength and direction of a linear relationship between two variables. It can take values between -1 and 1, where -1 indicates a perfect negative correlation, 0 indicates no correlation, and 1 indicates a perfect positive correlation.To calculate the correlation coefficient, we can use the following formula:$$r = \frac{\sum(x - \overline{x})(y - \overline{y})}{\sqrt{\sum(x - \overline{x})^2 \sum(y - \overline{y})^2}}$$where x and y are the two variables, and $\overline{x}$ and $\overline{y}$ are their respective means.Using the data given in the question, we can calculate the correlation coefficient as follows:$$\overline{x} = \frac{76 + 89 + 95 + 94 + 92 + 102 + 79 + 104 + 84}{9} = 90.11$$$$\overline{y} = \frac{12 + 16 + 19 + 19 + 17 + 24 + 13 + 24 + 14}{9} = 16.89$$$$\sum(x - \overline{x})(y - \overline{y}) = (76 - 90.11)(12 - 16.89) + (89 - 90.11)(16 - 16.89) + \cdots + (84 - 90.11)(14 - 16.89) = -108.93$$$$\sum(x - \overline{x})^2 = (76 - 90.11)^2 + (89 - 90.11)^2 + \cdots + (84 - 90.11)^2 = 1025.39$$$$\sum(y - \overline{y})^2 = (12 - 16.89)^2 + (16 - 16.89)^2 + \cdots + (14 - 16.89)^2 = 102.53$$Therefore, the correlation coefficient is:$$r = \frac{-108.93}{\sqrt{1025.39 	imes 102.53}} = -0.881$$The correct answer is C) 0.881.

The Data Below Are the Temperatures on Randomly Chosen Days