John sells frozen fruit bars at a stand in a park during the summer months. He records the average weekly temperature and the number of frozen fruit bars sold for 6 weeks.
To approach the problem of determining the correlation between temperature and the number of fruit bars sold, as well as understanding the real-world meaning of the slope of the line of best fit, we need to consider the following steps:
1. Determine the Correlation: The correlation coefficient measures the strength and direction of a linear relationship between two variables. The correlation coefficient ranges from -1 to 1: - A correlation close to 1 implies a strong positive correlation. - A correlation close to -1 implies a strong negative correlation. - A correlation close to 0 implies no linear correlation.
2. Calculate the Line of Best Fit: The line of best fit is determined through linear regression, which calculates the best-fitting line through the data points. The equation of this line is: [tex]\[
y = mx + b
\][/tex] where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
3. Interpret the Slope: The slope ([tex]\(m\)[/tex]) indicates the rate of change in the dependent variable (fruit bars sold) for each unit change in the independent variable (temperature).
Given the provided data for temperatures and the number of fruit bars sold: [tex]\[
\begin{array}{|c|c|}
\hline
\text{Temperature} \left({ }^{\circ} F \right) & \text{Fruit Bars Sold} \\
\hline
67 & 50 \\
\hline
71 & 54 \\
\hline
76 & 63 \\
\hline
76 & 65 \\
\hline
82 & 65 \\
\hline
87 & 100 \\
\hline
\end{array}
\][/tex]
The correlation coefficient is calculated to be: [tex]\[
0.8928304366620293
\][/tex] This value is close to 1, suggesting a strong positive correlation between temperature and the number of fruit bars sold.
The line of best fit has a slope ([tex]\(m\)[/tex]) of: [tex]\[
2.1854684512428295
\][/tex] and an intercept ([tex]\(b\)[/tex]) of: [tex]\[
-101.0216698534098
\][/tex]
### Real-World Meaning of the Slope The slope of the line, approximately [tex]\(2.185\)[/tex], represents the change in the number of fruit bars sold for every unit increase in temperature. Specifically, this means:
"There are approximately [tex]\(2.185\)[/tex] more fruit bars sold for every [tex]\(1\)[/tex] degree Fahrenheit the temperature rises."
To make this a complete sentence: There are approximately 2.185 more fruit bars sold for every 1 degree the temperature rises.
