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Anna set up a lemonade stand on her block over the summer. She recorded each day's high temperature and the number of cups of lemonade she sold for 10 days.

After plotting her results, Anna noticed that the relationship between her two variables was fairly linear. She used her data to calculate the following least squares regression equation for predicting lemonade sales, in cups, from the daily high temperature, in degrees Fahrenheit:

[tex]\ \textless \ br/\ \textgreater \ \hat{y} = -34 + \frac{8}{5} x\ \textless \ br/\ \textgreater \ [/tex]

What is the residual if a day had a high temperature of 95 degrees and Anna sold 21 cups of lemonade?



Answer :

GinnyAnswer
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.

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