Sure, let's go through the steps to determine the line of best fit for the given data, where [tex]\( x \)[/tex] represents the average daily temperature and [tex]\( y \)[/tex] represents the total ice cream sales. We aim to find a linear equation of the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
### Step-by-Step Solution
1. Gather the Data:
- Temperatures: 8.2, 64.2, 64.3, 66.8, 68.4, 71.6, 72.7, 76.2, 77.8, 82.8
- Sales: 112, 135, 138, 146, 166, 180, 182, 199, 220, 280
2. Calculate the Line of Best Fit:
- Calculate the slope ([tex]\( m \)[/tex]) and the y-intercept ([tex]\( b \)[/tex]) using linear regression techniques.
- Use the least squares method to minimize the difference between the observed values and the values predicted by the linear equation.
3. Resulting Line of Best Fit:
- Slope ([tex]\( m \)[/tex]): After performing the least squares regression, we get the slope as 1.3 (rounded to the nearest tenth).
- Y-intercept ([tex]\( b \)[/tex]): The y-intercept is calculated to be 74.5 (rounded to the nearest tenth).
4. Correlation Coefficient ([tex]\( r \)[/tex]):
- The correlation coefficient gives an idea of how well the line fits the data.
- Here, the correlation coefficient ([tex]\( r \)[/tex]) is approximately 0.391. This value indicates a moderate positive relationship between temperature and ice cream sales.
5. Statistical Values:
- P-value: The p-value of 0.26395 suggests that there is a moderate probability that the relationship observed is due to random chance.
- Standard Error: The standard error of the slope is around 1.083, indicating the variability in the estimate of the slope.
### Summary
The equation of the line of best fit, with values rounded to the nearest tenth, is:
[tex]\[ y = 1.3x + 74.5 \][/tex]
Where:
- [tex]\( x \)[/tex] is the average daily temperature.
- [tex]\( y \)[/tex] is the total ice cream sales.
This equation can be used to predict ice cream sales based on the average daily temperature. If you have a temperature value, you can plug it into this equation to estimate the expected sales.
