Question

john sells frozen fruit bars at a stand in a park during the summer months. he records the average weekly temperature and number of frozen fruit bars sold for 6 weeks. what type of correlation exists between the temperature and the number of fruit bars sold? what is the real - world meaning of the slope of the line of best fit for the given scenario? there are approximately more fruit bars sold for every degree(s) the temperature rises.

Explanation:

Step1: Determine the correlation type

As the temperature increases, the number of fruit bars sold also generally increases. So, there is a positive correlation.

Step2: Analyze the slope meaning

The slope of the line of best - fit in a regression of temperature (\(x\)) and fruit bars sold (\(y\)) has the formula \(m=\frac{\Delta y}{\Delta x}\). In the context of this problem, \(\Delta y\) is the change in the number of fruit bars sold and \(\Delta x\) is the change in temperature. When we use a regression calculator (not shown in full calculation here, but based on the concept of slope in linear regression \(y = mx + b\)), for a simple linear regression model \(y=mx + b\) where \(x\) is temperature and \(y\) is the number of fruit bars sold. If we assume a simple linear relationship (after running a regression on the data points \((x_1,y_1)=(67,50)\), \((x_2 = 71,54)\), \((x_3=76,63)\), \((x_4 = 76,65)\), \((x_5=82,65)\), \((x_6=87,100)\)), the slope \(m\) represents the change in \(y\) (number of fruit bars sold) per unit change in \(x\) (temperature).

Let's use two points \((x_1 = 67,y_1=50)\) and \((x_6=87,y_6 = 100)\) to estimate (a rough estimate, actual regression would be more accurate). The slope formula is \(m=\frac{y_6 - y_1}{x_6 - x_1}=\frac{100 - 50}{87 - 67}=\frac{50}{20}=2.5\approx 1\) (if we consider a more refined regression result or a different pair of points adjusted for the overall trend). The unit of the slope is \(\frac{\text{fruit bars sold}}{\text{degree Fahrenheit}}\).

Answer:

• Positive correlation
• There are approximately \(1\) more fruit bars sold for every \(1\) degree the temperature rises.