- Calculate the sums \(\sum x\), \(\sum y\), \(\sum xy\), and \(\sum x^2\) from the given data.
- Apply the least-squares regression formulas for slope \(m\) and intercept \(b\).
- Round the resulting coefficients to the nearest tenth to obtain the linear equation \(y = 6.5x - 279.1\).
- Final Answer: \(\boxed{y = 6.5x - 279.1}\)
Explanation
- Data and Problem Analysis
To find the line of best fit (linear regression equation) \(y = mx + b\) for the given data, we first analyze the set of 10 data points:
- Temperature (x): 58.2, 64.2, 64.3, 66.8, 68.4, 71.6, 72.7, 76.2, 77.8, 82.8
- Sales (y): 112, 135, 138, 146, 166, 180, 188, 199, 220, 280
The number of data points is \(n = 10\). To calculate the slope \(m\) and intercept \(b\), we need the following sums:
- \(\sum x = 58.2 + 64.2 + 64.3 + 66.8 + 68.4 + 71.6 + 72.7 + 76.2 + 77.8 + 82.8 = 703.0\)
- \(\sum y = 112 + 135 + 138 + 146 + 166 + 180 + 188 + 199 + 220 + 280 = 1764\)
- \(\sum xy = (58.2 \times 112) + (64.2 \times 135) + (64.3 \times 138) + (66.8 \times 146) + (68.4 \times 166) + (71.6 \times 180) + (72.7 \times 188) + (76.2 \times 199) + (77.8 \times 220) + (82.8 \times 280) = 127185.4\)
- \(\sum x^2 = 58.2^2 + 64.2^2 + 64.3^2 + 66.8^2 + 68.4^2 + 71.6^2 + 72.7^2 + 76.2^2 + 77.8^2 + 82.8^2 = 49911.14\)
- Calculation of the Regression Line Constants
The formulas for the least-squares regression line \(y = mx + b\) are:
\[m = \frac{n \sum xy - (\sum x)(\sum y)}{n \sum x^2 - (\sum x)^2}\]
\[b = \frac{\sum y - m \sum x}{n}\]
Substituting the sums calculated in the previous step:
- \(n \sum xy = 10 \times 127185.4 = 1271854\)
- \((\sum x)(\sum y) = 703.0 \times 1764 = 1240092\)
- \(n \sum x^2 = 10 \times 49911.14 = 499111.4\)
- \((\sum x)^2 = 703.0^2 = 494209\)
Now, calculate the slope \(m\):
\[m = \frac{1271854 - 1240092}{499111.4 - 494209} = \frac{31762}{4902.4} \approx 6.478867\]
Now, calculate the intercept \(b\):
\[b = \frac{1764 - (6.478867 \times 703.0)}{10} = \frac{1764 - 4554.6435}{10} = \frac{-2790.6435}{10} \approx -279.064\]
- Rounding and Final Result
Rounding the slope \(m\) and the intercept \(b\) to the nearest tenth:
- \(m \approx 6.5\)
- \(b \approx -279.1\)
The final regression equation is:
\[y = 6.5x - 279.1\]
Examples
Linear regression is commonly used in business to predict future sales based on historical data. For example, if an ice cream shop owner observes how temperature affects daily sales, they can use the linear equation to estimate potential revenue for the upcoming week based on weather forecasts. This allows them to better manage inventory, such as ordering the right amount of milk and sugar, and scheduling enough staff to meet the expected demand, ultimately maximizing profit and reducing waste.