Question

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, so 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:

y^= -34+3/5x

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

__ cups

Answer 1

During the summer, Krista noted both the number of customers who came to her lemonade stand each day and how much the temperature rose during the day while her stand was open. Based on the data, she concluded that there is a positive correlation between the number of customers and the increase in temperature. Identify which of the two data tables represents a positive correlation. Then, plot the set of data points from that table.

Table1

Increase in Temperature (°F) Number of Customers

10 10

8 9

11 12

15 17

15 16

16 18

11 11

16 13

14 15

13 14

Table2

Increase in Temperature (°F) Number of Customers

3 10

2 14

4 12

5 13

3 10

3 15

7 8

10 5

11 6

12 8

Explanation:

Answer 2

Residual = -2

The negative residual value indicates that the data point lies below the regression line.

Explanation:

We are given a linear regression model that relates daily high temperature, in degrees Fahrenheit and number of lemonade cups sold.

`y = -34+ (3)/(5) x`

Where y is the number of cups sold and x is the daily temperature in Fahrenheit.

Residual value:

A residual value basically shows the position of a data point with respect to the regression line.

A residual value of 0 is desired which means that the regression line best fits the data.

The Residual value is calculated by

Residual = Observed value - Predicted value

The predicted value of number of lemonade cups is obtained as

`y = -34+ (3)/(5) (95)\\\\y = -34+ 3 (19)\\\\y = -34+ 57\\\\y = 23`

So the predicted value of number of lemonade cups is 23 and the observed value is 21 so the residual value is

Residual = Observed value - Predicted value

Residual = 21 - 23

Residual = -2

The negative residual value indicates that the data point lies below the regression line.