To find the residual for the given data, we need to follow these steps:
1. Identify the given details:
- High temperature ([tex]\( x \)[/tex]) = 95 degrees Fahrenheit.
- Actual sales (number of cups sold) = 21 cups of lemonade.
- The least squares regression equation is given as [tex]\( \hat{y} = -34 + \frac{8}{5} x \)[/tex].
2. Calculate the predicted sales using the regression equation:
[tex]\[
\hat{y} = -34 + \frac{8}{5} \times 95
\][/tex]
First, calculate [tex]\( \frac{8}{5} \times 95 \)[/tex]:
[tex]\[
\frac{8}{5} \times 95 = \frac{8 \times 95}{5} = \frac{760}{5} = 152
\][/tex]
Then, use this result in the regression equation:
[tex]\[
\hat{y} = -34 + 152 = 118
\][/tex]
So, the predicted sales ([tex]\( \hat{y} \)[/tex]) is 118 cups of lemonade.
3. Calculate the residual:
The residual is the difference between the actual sales and the predicted sales:
[tex]\[
\text{Residual} = \text{Actual Sales} - \text{Predicted Sales}
\][/tex]
[tex]\[
\text{Residual} = 21 - 118 = -97
\][/tex]
Therefore, the predicted sales for a day with a high temperature of 95 degrees is 118 cups of lemonade, and the residual is -97.
