Answer :
To determine the slope of the line of best fit, where [tex]\( x \)[/tex] represents the average daily temperature and [tex]\( y \)[/tex] represents the ice cream sales, we can follow these steps:
1. Identify the Data Points:
- The given data points are:
[tex]\[ \begin{array}{|c|c|} \hline \text{Temperature} (^{\circ}F) & \text{Ice Cream Sales} (\$) \\ \hline 58.2 & 112 \\ \hline 64.2 & 135 \\ \hline 64.3 & 138 \\ \hline 66.8 & 146 \\ \hline 68.4 & 166 \\ \hline 71.6 & 180 \\ \hline 72.7 & 188 \\ \hline 76.2 & 199 \\ \hline 77.8 & 220 \\ \hline 82.8 & 280 \\ \hline \end{array} \][/tex]
2. Formulate the Linear Model:
- We are looking for a linear relationship of the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
3. Compute the Slope [tex]\( m \)[/tex] of the Line of Best Fit:
- The slope [tex]\( m \)[/tex] can be determined from the formula for the line of best fit using methods like least squares regression. The formula for the slope [tex]\( m \)[/tex] is:
[tex]\[ m = \frac{N \sum (xy) - \sum x \sum y}{N \sum (x^2) - (\sum x)^2} \][/tex]
- Where [tex]\( N \)[/tex] is the number of data points, [tex]\(\sum (xy)\)[/tex] is the sum of the products of the paired data points, [tex]\(\sum x\)[/tex] and [tex]\(\sum y\)[/tex] are the sums of the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values respectively, and [tex]\(\sum (x^2)\)[/tex] is the sum of squares of the [tex]\( x \)[/tex] values.
4. Round the Slope to One Decimal Place:
- After calculation, we round the slope to one decimal place.
Given these calculations, the slope [tex]\( m \)[/tex] of the line of best fit turns out to be:
[tex]\[ m = 6.5 \][/tex]
Thus, the slope of the line of best fit, rounded to one decimal place, is [tex]\( 6.5 \)[/tex]. Therefore, the correct answer is:
[tex]\[ \boxed{6.5} \][/tex]
1. Identify the Data Points:
- The given data points are:
[tex]\[ \begin{array}{|c|c|} \hline \text{Temperature} (^{\circ}F) & \text{Ice Cream Sales} (\$) \\ \hline 58.2 & 112 \\ \hline 64.2 & 135 \\ \hline 64.3 & 138 \\ \hline 66.8 & 146 \\ \hline 68.4 & 166 \\ \hline 71.6 & 180 \\ \hline 72.7 & 188 \\ \hline 76.2 & 199 \\ \hline 77.8 & 220 \\ \hline 82.8 & 280 \\ \hline \end{array} \][/tex]
2. Formulate the Linear Model:
- We are looking for a linear relationship of the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
3. Compute the Slope [tex]\( m \)[/tex] of the Line of Best Fit:
- The slope [tex]\( m \)[/tex] can be determined from the formula for the line of best fit using methods like least squares regression. The formula for the slope [tex]\( m \)[/tex] is:
[tex]\[ m = \frac{N \sum (xy) - \sum x \sum y}{N \sum (x^2) - (\sum x)^2} \][/tex]
- Where [tex]\( N \)[/tex] is the number of data points, [tex]\(\sum (xy)\)[/tex] is the sum of the products of the paired data points, [tex]\(\sum x\)[/tex] and [tex]\(\sum y\)[/tex] are the sums of the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values respectively, and [tex]\(\sum (x^2)\)[/tex] is the sum of squares of the [tex]\( x \)[/tex] values.
4. Round the Slope to One Decimal Place:
- After calculation, we round the slope to one decimal place.
Given these calculations, the slope [tex]\( m \)[/tex] of the line of best fit turns out to be:
[tex]\[ m = 6.5 \][/tex]
Thus, the slope of the line of best fit, rounded to one decimal place, is [tex]\( 6.5 \)[/tex]. Therefore, the correct answer is:
[tex]\[ \boxed{6.5} \][/tex]
