Question

uses objective j

1. jennifer and barry run an ice cream shop. they gather data for a week about the number of cones they sell and the daily high temperature in order to better plan for sales, as shown at the right.

a. graph the data at the right.
b. write the equation of a regression line for the data.
c. find the slope of the regression line and explain what it means in the context
d. how many cones does the equation predict that they will sell on a day with a high temperature of 85°f?

Explanation:

Step1: Graphing the data (a)

• Plot each pair \((x,y)\) where \(x\) is the temperature (\(^{\circ}F\)) and \(y\) is the number of cones. For example, for the pair \((84,1520)\), locate \(x = 84\) on the temperature axis and \(y=1520\) on the cones axis and mark the point. Do this for all pairs \((84,1520),(81,1375),(87,1974),(79,920),(77,870),(74,643),(81,1497)\).

Step2: Finding the regression line (b)

• Using a calculator or statistical software (e.g., in Excel: Data - Data Analysis - Regression; in a TI - 84: Stat - Edit (enter data), then Stat - Calc - LinReg(ax + b)), input the \(x\) (temperature) and \(y\) (cones) values.
• Let's assume we use the formula for the least - squares regression line \(y=ax + b\), where \(a=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}\) and \(b=\overline{y}-a\overline{x}\), \(n = 7\) (number of data points).
• \(\sum_{i=1}^{7}x_i=84 + 81+87+79+77+74+81=563\), \(\sum_{i = 1}^{7}y_i=1520+1375 + 1974+920+870+643+1497=8809\), \(\sum_{i=1}^{7}x_i^{2}=84^{2}+81^{2}+87^{2}+79^{2}+77^{2}+74^{2}+81^{2}=45393\), \(\sum_{i = 1}^{7}x_iy_i=84\times1520+81\times1375+87\times1974+79\times920+77\times870+74\times643+81\times1497=737790\)
• \(a=\frac{7\times737790 - 563\times8809}{7\times45393-(563)^{2}}\approx107.7\), \(\overline{x}=\frac{563}{7}\approx80.43\), \(\overline{y}=\frac{8809}{7}\approx1258.43\), \(b = 1258.43-107.7\times80.43\approx - 7333.7\)
• The regression line is \(y = 107.7x-7333.7\)

Step3: Interpreting the slope (c)

• The slope \(a = 107.7\). In the context of the problem, it means that for every \(1^{\circ}F\) increase in the daily high temperature, the number of ice - cream cones sold is predicted to increase by approximately \(107.7\) cones.

Step4: Predicting the number of cones (d)

• Substitute \(x = 85\) into the regression equation \(y=107.7x - 7333.7\)
• \(y=107.7\times85-7333.7\)
• \(y = 9154.5-7333.7=1820.8\approx1821\)

Answer:

a. Plot the points \((84,1520),(81,1375),(87,1974),(79,920),(77,870),(74,643),(81,1497)\) on the given graph.
b. \(y = 107.7x-7333.7\)
c. The slope is \(107.7\). It means that for every \(1^{\circ}F\) increase in temperature, the number of cones sold is predicted to increase by about \(107.7\) cones.
d. Approximately \(1821\) cones.